# The General Linear Group as a basis for all nxn matrices

I'm trying to prove that every nxn matrix can be written as a linear combination of matrices in GL(n,F).

I know all matrices in GL(n,F) are invertible and hence have linearly independent columns and rows. I was thinking perhaps there is something about the joint bases for the n-dimensional column and row spaces, respectively, that could provide a basis for M_{nxn}(F), which has dimension of n^2.

Is this on the right track?