On the vanishing of subspaces of alternating bilinear forms

Given a field F and integer n ≥ 3, we introduce an invariant s _n(F) which is defined by examining the vanishing of subspaces of alternating bilinear forms on 2-dimensional subspaces of vector spaces. This invariant arises when we calculate the largest dimension of a subspace of n × n skew-symmetric matrices over F which contains no elements of rank 2. We show how to calculate s_n(F) for various families of field F, including finite fields. We also prove the existence of large subgroups of the commutator subgroup of certain p-groups of class 2 which contain no non-identity commutators.