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?
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.