Consider the Yang-Mills action $S = -\frac{1}{2g^{2}} \int d^{x}\ \mathrm{Tr}\left( F_{\mu \nu} F^{\mu \nu} \right)$, where $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\mu} A_{\nu} - i [A_{\mu},A_{\nu}]$ is the field strength tensor. Let the gauge group be $U(N)$, so that $A_{\mu}$ belongs to the adjoint representation of $U(N)$.
I'm trying to do a problem which says the following: the adjoint representation of $U(N)$ can be thought of as products of fundamental and anti-fundamental representations - so $\bar{N} \times N$. They say, adopt the following way to write the group index of $A_{\mu}^{a}$, such that $a = (\bar{j}, k)$. In this way, the above means the following in terms of matrix elements: $$ A^{a}_{\mu} \ \to \ A^{(\bar{j},k)}_{\mu} = \left[ A_{\mu} \right]_{jk} $$
I'm asked to compute the structure constants $f^{(\bar{m},n)}_{(\bar{j},k),(\bar{p},q)}$ which are defined by the expression: $$ [ A_{\mu}, A_{\nu} ]^{(\bar{m}, n)} = i f^{(\bar{m},n)}_{(\bar{j},k),(\bar{p},q)} A^{(\bar{j},k)}_{\mu} A^{(\bar{p},q)}_{\nu} $$
What exactly am I trying to do here?
My Attempt: My thinking is that I pick some set of generators $\{ T^{a} \}$ for $U(N)$, which defines a set of structure constants $\{f_{abc}\}$ (relative to my choice of generators).
The generators of the adjoint representation $\{T_{\mathrm{AD}}^{a}\}$, have elements determined by $[T_{\mathrm{AD}}^{a}]_{bc} = i f_{abc}$. Then I think I can expand $A_{\mu}$ as: $$ A_{\mu} = \sum _{a} A_{\mu}^{a} T_{\mathrm{AD}}^{a} $$
But from here I don't have any idea what i am doing....am I supposed to write the structure constants $f^{(\bar{m},n)}_{(\bar{j},k),(\bar{p},q)}$ in terms of the $f_{abc}$?
(Eventually this problem is to lead me towards the t'Hooft double line formalism.)
EDIT: It occurred to me that maybe I need to write the $f^{(\bar{m},n)}_{(\bar{j},k),(\bar{p},q)}$ in terms of the structure constants of the fundamental and anti-fundamental respresentations?