  We consider $q$-matroids and their associated classical matroids derived from
Gabidulin rank-metric codes. We express the generalized rank weights of a
Gabidulin rank-metric code in terms of Betti numbers of the dual classical
matroid associated to the $q$-matroid corresponding to the code. In our main
result, we show how these Betti numbers and their elongations determine the
generalized weight polynomials for $q$-matorids, in particular, for the
Gabidulin rank-metric codes. In addition, we demonstrate how the weight
distribution and higher weight spectra of such codes can be determined directly
from the associated $q$-matroids by using M\"{o}bius functions of its lattice
of $q$-flats.
