Suppose $G$ and $A$ are abelian groups (I'm setting $G$ abelian to keep the discussion simple, though there are analogues for non-abelian $G$) with $G$ acting trivially on $A$. By the universal coefficients theorem, we have short exact sequences for all positive integers $n$:

$$0 \to \operatorname{Ext}^1(H_{n-1}(G;\mathbb{Z});A) \to H^n(G;A) \to \operatorname{Hom}(H_n(G;\mathbb{Z}),A) \to 0$$

In the case $n = 2$, this becomes:

$$0 \to \operatorname{Ext}^1(G;A) \to H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0$$

The surjection $H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A)$ has a nice interpretation:

$H_2(G;\mathbb{Z})$ is the Schur multiplier of $G$, which (since $G$ is abelian) is the exterior square of $G$, so $\operatorname{Hom}(H_2(G;\mathbb{Z}),A)$ is the alternating bilinear maps from $G$ to $A$. The mapping $H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A)$ is the "skew" mapping; it sends a 2-cocycle $f:G \times G \to A$ to the function:

$$(x,y) \mapsto f(x,y) - f(y,x)$$

Even without prior knowledge of the universal coefficients theorem, this short exact sequence makes sense: one can check that the skew of a 2-cocycle for an abelian group is alternating and bilinear, and the mapping descends to cohomology classes because any 2-coboundary is symmetric. The kernel of the mapping corresponds to symmetric cohomology classes, which corresponding to the abelian extension groups, given precisely a $\operatorname{Ext}^1(G;A)$. In group extension terms, the image of a given element of $H^2(G;A)$ under the skew map describes the commutator map of the extension group.

My question is: what's an analogous concrete interpretation for higher $n$?

I'm listing below some references with generalizations of the above to non-abelian $G$ and varietal generalizations:

John Burns and Graham Ellis. On the nilpotent multipliers of a group. Math. Zeitschr., Volume 226, Page 405 - 428(Year 1997) (Official gated copy (PDF))

C. R. Leedham-Green and Susan McKay. Baer-invariants, isologism, varietal laws, and homology. PDF online. Pages 37-41 (135-139 in the original) describe a generalization of universal coefficients to varietal laws (where the usual universal coefficients is with respect to abelian groups inside groups)