We show that self-dual two-forms in 2n-dimensional spaces determine a n(2)-n+1-dimensional manifold S-2n and the dimension of the maximal linear subspaces of S-2n is equal To the (Radon-Hurwitz) number of linearly independent vector fields on the sphere S2n-1. We provide a direct proof that for n odd S-2n has only one-dimensional linear submanifolds. We exhibit 2(c)-1-dimensional subspaces in dimensions which are multiples of 2(c), for c=1,2,3. In particular, we demonstrate that the seven-dimensional linear subspaces of S-8 also include among many other interesting classes of self-dual two-forms, the self-dual two-forms of Corrigan, Devchand, Fairlie, and Nuyts [Nucl. Phys. B 214, 452 (1983)] and a representation of Cl-7 given by octonionic multiplication. We discuss the relation of the Linear subspaces with the representations of Clifford algebras. (C) 1997 American Institute of Physics.