  We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$
dimensional manifold ${\cal S}_{2n}$ and the dimension of the maximal linear
subspaces of ${\cal S}_{2n}$ is equal to the (Radon-Hurwitz) number of linearly
independent vector fields on the sphere $S^{2n-1}$. We provide a direct proof
that for $n$ odd ${\cal S}_{2n}$ 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 ${\cal S}_{8}$ also include among many other interesting
classes of self-dual 2-forms, the self-dual 2-forms of Corrigan, Devchand,
Fairlie and Nuyts and a representation of ${\cal C}l_7$
  given by octonionic multiplication. We discuss the relation of the linear
subspaces with the representations of Clifford algebras.
