Lie algebra of $GL(n)$

Is the set of all n-by-y square matrices.

Because $GL(n)$ is a Lie group we can use Section "Lie algebra of a matrix Lie group".

For every matrix $x$ in the set of all n-by-y square matrices $M_n$, $e^x$ has inverse $e^{-}x$.

Note that this works even if $x$ is not invertible, and therefore not in $GL(n)$!

Therefore, the Lie algebra of $GL(n)$ is the entire $M_n$.