I used the code provided by Andrew J, but the DataType CellValues.Date produced a corrupted xlsx-file for me.
The DataType CellValues.Number worked fine for me (Don't forget to set NumberFormatId):
cell.DataType = new EnumValue<CellValues>(CellValues.Number);
My whole code:
DateTime valueDate = DateTime.Now;
string valueString = valueDate.ToOADate().ToString();
CellValue cellValue = new CellValue(valueString);
Cell cell = new Cell();
cell.DataType = new EnumValue<CellValues>(CellValues.Number);
cell.StyleIndex = yourStyle; //StyleIndex of CellFormat cfBaseDate -> See below
cell.Append(cellValue);
My CellFormat for this cell in the Stylesheet looks like:
CellFormat cfBaseDate = new CellFormat() { 
 ApplyNumberFormat = true,
 NumberFormatId = 14, //14 is a localized short Date (d/m/yyyy) -> See list below
 //Some further styling parameters
}; 
If you'd like to format your date another way, here is a list of all default Excel NumberFormatId's
ID  FORMAT CODE
0   General
1   0
2   0.00
3   #,##0
4   #,##0.00
9   0%
10  0.00%
11  0.00E+00
12  # ?/?
13  # ??/??
14  d/m/yyyy
15  d-mmm-yy
16  d-mmm
17  mmm-yy
18  h:mm tt
19  h:mm:ss tt
20  H:mm
21  H:mm:ss
22  m/d/yyyy H:mm
37  #,##0 ;(#,##0)
38  #,##0 ;[Red](#,##0)
39  #,##0.00;(#,##0.00)
40  #,##0.00;[Red](#,##0.00)
45  mm:ss
46  [h]:mm:ss
47  mmss.0
48  ##0.0E+0
49  @
Source of list: https://github.com/ClosedXML/ClosedXML/wiki/NumberFormatId-Lookup-Table
I know this list is from ClosedXML, but it's the same in OpenXML.