Is there a formula for the area of a general polytope with $n$ vertices which just uses the distances between vertices, like Herons formel for the area of a triangle?
In general, the area of an $n$-gon cannot be deduced from the distances between vertices alone. For example, the two $8$-gons
have the vertices in the same places (and thus the distances between vertices will be the same), but will have different areas.
Your question mentions polytopes, so you could be asking whether Heron's formula generalizes to higher $n$-dimensional simplices. For such a simplex, every pair of points is connected by an edge, and unlike the context of Rebecca's answer, the resulting shape is uniquely determined by all of these distances. There is indeed a more general formula, given quite elegantly by Cayley-Menger determinants.