We study the orbits of vector spaces of skew-symmetric matrices of constant rank 2r and type (N+1)×(N+1) under the natural action of SL(N+1), over an algebraically closed field of characteristic zero. We give a complete description of the orbits for vector spaces of dimension 2, relating them to some 1-generic matrices of linear forms. We also show that, for each rank two vector bundle on P^2 defining a triple Veronese embedding of P^2 in G(1,7), there exists a vector space of 8×8 skew-symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle.
