First, you are mixing the mysql api's at somepoint you are using mysqli_* at some point u using mysql_* They don't mix. And mysql_* functions are depreciated they no longer supported by later versions of php. better use mysqli or pdo. this mysql_real_escape_string() or  mysqlo_real_escape_string() is not safe enough to prevent you against sql injections. solution is simple better start using mysqli prepared statements or pdo prepared statements.
another error : <input type="text" name="name"> <input type="text" name="name">  these two inputs fields have the same name attribute php will only read one. and you will get an undefined index here $misure      = $_POST['misure'];  You need to activate error reporting while you are still developing so you can see your errors and notices:
add this at the top of every php page : ini_set('display_errors', 1); 
error_reporting(E_ALL);
also date date is a reserved word for mysql so you better use something else for your column name or add backslashes date
Oh and your code never execute here :
if (isset($_POST['submit']))
 { 
 // get form data, making sure it is valid
 $name = mysql_real_escape_string(htmlspecialchars($_POST['name']));
 $author = mysql_real_escape_string(htmlspecialchars($_POST['author']));
  $description = mysql_real_escape_string(htmlspecialchars($_POST['description']));
 $misure = mysql_real_escape_string(htmlspecialchars($_POST['misure']));
 $date = mysql_real_escape_string(htmlspecialchars($_POST['date']));
  $status = mysql_real_escape_string(htmlspecialchars($_POST['status']));
 }
Why is that? because you do not have POST value with the submit attribute name. <input type="submit"> see? your submit does not have a name attribute. therefore. This means
all this :
VALUES ('$name', '$author', '$description', '$misure', '$date', '$status')"; These are all undefined variables. I'm surprised why doesn't your server tell you that, with that error reporting enable you will get all those.
This is what u need to do to solve that : 
Your html side.
<form action="uploadall.php" method="post">
Name: <input type="text" name="name"><br>
Autore: <input type="text" name="author"><br>
Descrizione: <textarea id="editordescription" name="description" cols="45" rows="15">
        </textarea>
        <script>
            CKEDITOR.replace( 'editordescription' );
        </script>
<br>Misure: <input type="text" name="misure"><br>
Data: <input type="text" name="date"><br>
    <input type="hidden" name="status" value="Disattivo" size="20">
<input type="submit" name="submit">
</form>
uploadall.php
<?php
// check if the form has been submitted. If it has, start to process the form and save it to the database
if (isset($_POST['submit'])) {
    $servername = "xxxxxxx";
    $username   = "xxxxxxx";
    $password   = "xxxxxxx";
    $dbname     = "xxxxxxxxx";
    // Create connection
    $conn = new mysqli($servername, $username, $password, $dbname);
    // Check connection
    if ($conn->connect_error) {
        die("Connection failed: " . $conn->connect_error);
    }
    //check your inputs are set and validate,filter and sanitize
    $name        = $_POST['name'];
    $author      = $_POST['author'];
    $description = $_POST['description'];
    $misure      = $_POST['misure'];
    $date        = $_POST['date'];
    $status      = $_POST['status'];
    //prepare and bind
    $sql = $conn->prepare("INSERT INTO exposition (name, author, description, misure, date, status)
VALUES (?,?,?,?,?,?)");
    $sql->bind_param("ssssss", $name, $author, $description, $misure, $date);
    if ($sql->execute()) {
        echo "New record created successfully";
    } else {
        //you have an error
    }
    $conn->close();
}
?>
That's all good luck.
Update :
I corrected errors you told me and I am using PDO now but it still
  doesn't work
I read that from your comments above, but you not telling us what the errors are, but I believe they are the ones I highlighted above.
with PDO this is how u will achieve your goal :
<?php
    //connection
    $servername = 'XXXXXXXXXXXXX';
    $dbname     = 'XXXXXXXXXXXXX';
    $username   = 'XXXXXXXXXXXXXX';
    $password   = 'XXXXXXXXX';
    $charset    = 'utf8';
    $dsn = "mysql:host=$servername;dbname=$dbname;charset=$charset";
    $opt = [
            PDO::ATTR_ERRMODE            => PDO::ERRMODE_EXCEPTION,
            PDO::ATTR_DEFAULT_FETCH_MODE => PDO::FETCH_ASSOC,
            PDO::ATTR_EMULATE_PREPARES   => false,
            ];
    $dbh = new PDO($dsn, $username, $password, $opt);
// check if the form has been submitted. If it has, start to process the form and save it to the database
if (isset($_POST['submit'])) {
    //check your inputs are set and validate,filter and sanitize
    $name        = $_POST['name'];
    $author      = $_POST['author'];
    $description = $_POST['description'];
    $misure      = $_POST['misure'];
    $date        = $_POST['date'];
    $status      = $_POST['status'];
    //prepare and bind
    $stmt = $dbh->prepare("INSERT INTO exposition (name, author, description, misure, date, status)VALUES (?,?,?,?,?,?)");
    if ($stmt->execute(array($name,$author,$description,$misure,$date,$status))) {
        echo "New Record inserted success";
    }
}
?>