Let $V$ be a vector space with a basis $B=\{e_1,\dots,e_n\}$, and let $F:V\to V$ be a linear map. The matrix of $F$ in the basis $B$ is given by
$$ [F]_B = ([F(e_1)]_B,\dots,[F(e_n)]_B) $$
But if I consider for example the vector space $M_{mn}(\mathbb{C})$ of $m\times n$ matrices under matrix addition, in which the standard basis is given by $B = \{E_{ij} : 1\leq i \leq m, \ 1\leq j \leq n\}$, where $E_{ij}$ is the $m\times n$ matrix with zeros everywhere except in position $ij$ where it is $1$, then how would I go about writing out the matrix for a linear operator $F:M_{mn}(\mathbb{C})\to M_{mn}(\mathbb{C})$? Would it be
$$ [F]_B = ([F(E_{11})]_B,[F(E_{21})]_B,\dots,[F(E_{m1})]_B,[F(E_{12})]_B,[F(E_{22})]_B,\dots,[F(E_{mn})]_B), $$
or
$$ [F]_B = ([F(E_{11})]_B,[F(E_{12})]_B,\dots,[F(E_{1n})]_B,[F(E_{21})]_B,[F(E_{22})]_B,\dots,[F(E_{mn})]_B), $$
or is some other way preferable? I think it probably does not matter as long as one keeps track of which coordinate corresponds to which $E_{ij}$. But I am not sure, and I would like to know if there is some standard way of doing it.
Exactly. What is really important is the operator action on the basis elements, in your case the action of $F$ on the $E_{ij}$. The matrix representation for an operator in a given basis does depend on the basis ordering, a permutation in the basis ordering will correspond to a column permutation for the operator matrix representation and you only need to keep track of it.