Introduction:
A variant of the Poincaré Lemma states that given a closed holomorphic $p$-form $\omega$ on a star-shaped $U \subset \mathbb C^n$, then there is a $(p-1)$-form $\varphi$ such that $\omega = d\varphi$. The proof of Poincaré's Lemma which I know provides us with a constructive way to find a primitive $\varphi$. For instance, take the $2$-form $\omega = x\cdot dy \wedge dz - y\cdot dz \wedge dx$ on $\mathbb C^3$, which is closed. $\varphi$ is then given by $\varphi = -\frac13 zy \cdot dx - \frac13 xz\cdot dy + \frac23 xy\cdot dz$.
My question:
I would like to know if there is a similar way of finding such "primitives" in group cohomology, in the following sense. Let $G$ be a group and $A$ a $G$-module. Suppose one has a function $h \colon G^n \to A$ which satisfies $dh = 0$ and its class $[h]$ is trivial in $H^n(G,A)$. Then there is some $f \colon G^{n-1} \to A$ with $df = h$. Is there any constructive way to find such an $f$?