I am starting a study of Lie Algebras and the text I am working out of simply states the following. I have been trying to come up with a proof but I am currently at a loss.
Let $V$ be a vector space of dimension $n$. We denote $L(End(V))$ as $gl(V)$. gl(V) is isomorphic to $gl(n)=L(M_n(k))$ of $n\times n$ matrices with entries in $k$.
The context here is that for any associative algebra $A$ we can define a lie bracket as $[a,b]=ab-ba$ for all $a,b\in A$. This give $A$ a Lie structure and we denote the Lie algebra as $L(A)$. Also $k$ is just some base field usually thought of as $\mathbb{C}$.
Let $B$ be a basis of $V$. For each $f\in\operatorname{End}V$, let $M_f=[f]_B^B$. Then$$\begin{array}{ccc}\mathfrak{gl}(V)&\longrightarrow&\mathfrak{gl}(n,k)\\f&\mapsto&M_f\end{array}$$is a Lie algebra isomorphism.