Probably the right thing to do is to express the classifying space of $A_n$ as the non-trivial double cover of the classifying space of $S_n$.  A point in the classifying space is then a set of $n$ points in $\mathbb{R}^n$ with a "sign ordering".  A sign ordering is an equivalence class of orderings of the points, i.e., ways to number them from 1 to $n$, up to even permutations.  I coined the term "sign ordering" by analogy with a cyclic ordering. But that name aside, the idea comes up all the time in various guises.  For instance an orientation of a simplex is by definition a sign ordering of its vertices.

This is in the same vein as your other examples and you can of course do something similar with any subgroup $G \subseteq S_n$.  You can always choose an ordering of the points up to relabeling by an element of $G$.