Let $G$ be a group which acts linearly on a vector space of dimension $n$ over a field $k$. Denote by $\rho$ this representation and consider the associated adjoint representation $Ad\rho$ which is just the representation of $G$ on $M_n(k)$ where $G$ acts by conjugation via $\rho$.
Now, working with group cohomology, we have the cup product $\cup : H^1(G,Ad \rho) \times H^1(G,Ad \rho) \to H^2(G,Ad\rho\otimes_{\mathbb{Z}}Ad\rho)$.
If I am not mistaken, this cup product is skew symmetric, hence if $u$ denotes the class of a 1-cocycle in $H^1(G,A\rho)$, then $2(u \cup u )= 0 \in H^2(G,Ad\rho\otimes_{\mathbb{Z}}Ad\rho)$.
Can we find an "explicit" function $f : G \to M_n(k)\otimes M_n(k)$ such that for all $\sigma,\tau \in G$, $2(u \cup u) (\sigma,\tau) = u(\sigma)\otimes \rho(\sigma)u(\tau)\rho(\sigma)^{-1} = \sigma.f(\tau) - f(\sigma \tau) + f(\sigma)$ ?