Isomorphism between $\mathbb{gl}(V)$ and $\mathbb{gl}(n,\mathbb{F})$ if $V\cong \mathbb{F}^n$ This is probably a standard example, but I couldn't really find it anywehere and I'm unsure if I understood it correctly... Take any finite dimensional vectorspace $V\cong \mathbb{F}^n$, then the Lie algebra $\mathfrak{gl}(V)$ with the commutator as Lie bracket is nothing elese than $\operatorname{End}(V)$. Since $V$ is finite dimensional we can represented the linear transformations as matrices, so we can write down $\mathfrak{gl}(n,\mathbb{F})$ and take the Lie bracket to be the commutator again. I think we have $\mathfrak{gl}(V)\cong \mathfrak{gl}(n,\mathbb{F})$. Both have dimension $n^2$ (for finite $n$) so by some theorem of linear algebra they are isomorphic. 
What bothers me a bit is that I'm struggeling to write down an explicit isomorphisom of Lie algebras. For $\mathfrak{gl}(n,\mathbb{F})$ one can just take the basis $\{e_{ij}\}_{1\le i,j\le n}$ of matrices with $1$ at position $(i,j)$ and zeros eleswhere. The basis for $\mathfrak{gl}(V)$ I think could be chosen as the set of maps $f_{i,j}:V\to V$ and $b_k \mapsto \delta_{ik}b_j$ if $V=\operatorname{span}(b_1, \dots, b_n)$. What I'm looking for is a map $\varphi$, such that 
$$\begin{align*}
\varphi: \mathfrak{gl}(V)&\longrightarrow \mathfrak{gl}(n,\mathbb{F}) 
\end{align*} 
$$
and $\varphi([f_{i,j},f_{k,l}]) = [\varphi(f_{i,j}),\varphi(f_{k,l})]$, but I don't really see what the right way is to map $f_{i,j}$ onto $e_{i,j}$. One could of course just set $\varphi(f_{i,j})(b_k) = e_{ij}b_k$, but I'm struggeling to show that this satisfies the homomorphisam property...
Am I making some very fundamental mistake here that I'm overlooking?
 A: You need much more than the two objects being isomorphic as vector spaces. What you need here is the correspondence between linear maps and matrices.  Given vector spaces $V$ and $W$ with bases $v_1, \ldots v_m$ and $w_1,\ldots, w_n$, and given a linear map $T:V\to W$, linearity means that the value of $T$ on any $v$ will be determined by its value on each $v_i$, and $T(v_j)=\sum a_{ij} w_i$.  If we assign to $T$ the matrix $[T]$ such that $[T]_{ij}=a_{ij}$, then the correspondence from linear maps between vector spaces with bases and matrices sends composition of linear maps to matrix multiplication.  In particular, $GL_n(\mathbb F)\cong GL(\mathbb F^n)$ as $\mathbb F$-algebras, and so under the bracket $[A,B]=AB-BA$, you get isomorphic Lie algebras.  In other words, $\mathfrak{gl}_n{\mathbb F}\cong \mathfrak{gl}(\mathbb F^n)$.
The explicit isomorphism between the Lie algebras is exactly the isomorphism between linear maps and matrices, but it requires specifying a basis first.  If you take the standard basis for $\mathbb F^n$, then you have a basis for $GL(V)$ of the form $T_{k\ell}(e_i)=\delta_{ik}e_{\ell}$.  These maps correspond to matrices with all but a single entry equal to 0 (and that last entry equal to 1).  While this gives you a set of generators and relations for the Lie algebra, and while  this is useful for certain computations, it isn't terribly useful for understanding the isomorphism.
