Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the cup-product.
Note. We take the tensor product $n$-times.
Question. How should I think about this map?
Note. By space I mean: topological space, manifold, analytic variety, algebraic variety, etc.
The cohomology I use should be a "useful" cohomology.
Is there any fundamental group hidden in this map?