The Chow ring of a matroid (or more generally, atomic latice) is an invariant
whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to
resolve the long-standing Heron-Rota-Welsh conjecture. Here, we make a detailed
study of the Chow rings of uniform matroids and of matroids of finite vector
spaces. In particular, we express the Hilbert series of such matroids in terms
of permutation statistics; in the full rank case, our formula yields the
maj-exc q-Eulerian polynomials of Shareshian and Wachs. We also provide a
formula for the Charney-Davis quantities of such matroids, which can be
expressed in terms of either determinants or q-secant numbers