You haven't provided nor a reproducible example, nor the desired output, so I'll have to guess
If this is your column names vector
vec <- LETTERS[1:3]
And this is your data set
set.seed(1)
df <- data.frame(A = sample(10, 10), 
                 B = sample(20, 10),
                 C = sample(30, 10))
Then you can try something like
lapply(vec, 
       function(x) lm(as.formula(paste(x, "~", 
                                       paste(setdiff(names(df), x), 
                                                     collapse = "+"))), 
                      data = df))
Which will give
# [[1]]
# 
# Call:
#   lm(formula = as.formula(paste(x, "~", paste(setdiff(names(df), 
#                                                       x), collapse = "+"))), data = df)
# 
# Coefficients:
#   (Intercept)            B            C  
# 4.9687       0.2410      -0.1565  
# 
# 
# [[2]]
# 
# Call:
#   lm(formula = as.formula(paste(x, "~", paste(setdiff(names(df), 
#                                                       x), collapse = "+"))), data = df)
# 
# Coefficients:
#   (Intercept)            A            C  
# 2.7975       0.8182       0.2775  
# 
# 
# [[3]]
# 
# Call:
#   lm(formula = as.formula(paste(x, "~", paste(setdiff(names(df), 
#                                                       x), collapse = "+"))), data = df)
# 
# Coefficients:
#   (Intercept)            A            B  
# 13.200       -1.675        0.875