  Moduli spaces of complete skew-forms are compactifications of spaces of
skew-symmetric linear maps of maximal rank on a fixed vector space, where the
added boundary divisor is simple normal crossing. In this paper we compute
their effective, nef and movable cones, the generators of their Cox rings, and
for those spaces having Picard rank two we give an explicit presentation of the
Cox ring. Furthermore, we give a complete description of both the Mori chamber
and stable base locus decompositions of the effective cone of some spaces of
complete skew-forms having Picard rank at most four.
