While I do not know any complete description, there are several quite general results in Section V.6 of the book Cohomology of groups of K. S. Brown. Here is a summary. In what follows, $R$ is a commutative ring with unit and $G$ is an abelian group.
There is a graded homomorphism $ \psi \colon \Lambda^*(G \otimes R) \to H_*(G;R)$ which is natural in $G$.
The map $\psi$ is always an isomorphism in degrees $0$ and $1$. If $R$ is a principal ideal domain of characteristic $0$ (meaning its fraction field has characteristic $0$), then it is also an isomorphism in degree $2$.
If $R$ is a principal ideal domain, $\psi$ is injective. Moreover, if $G$ is finitely generated it is a split injection.
If $R$ is a principal ideal domain and every prime $p$ such that $G$ has $p$-torsion is invertible in $R$, then $\psi$ is an isomorphism. In particular, it is an isomorphism if:
a) $ R = \mathbb{Q}$
b) $ R = \mathbb{Z}/p$ and $G$ is $p$-torsion-free.
c) $ R = \mathbb{Z}$ and $G$ is torsion-free.
Finally, it is also shown that if $G$ has $p$-torsion, then $$H_*(G;\mathbb{Z}/p) \cong \Lambda^*(G \otimes \mathbb{Z}/p) \otimes \Gamma(\text{Tor}(G,\mathbb{Z}/p))$$
where $\Gamma(S)$ denotes the divided polynomial algebra over $S$.