Smooth maps into quaternionic Grassmannians inducing a prescribed 4-form

Let knGk(Hn) be the Stiefel bundle of quaternionic k-frames in Hn over the the quaternionic Grassmannian Gk(Hn). Let sigma denote the first symplectic Pontrjagin form associated with the universal connection on k We show that every 4-form on a smooth manifold M can be induced from sigma by a smooth map f:MGk(Hn) (for sufficiently large k and n) provided there exists a continuous map Gk(Hn) which pulls back the deRham cohomology class of sigma (referred as the symplectic Pontrjagin class of kn) onto that of omega.