  The geometry of self-dual 2-forms in eight dimensions is studied. These
2-forms determine an $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$. We prove
that for add $n$, it has only one dimensionallinear subspaces. In eight
dimensions, the self-dual forms of Corrigan et al constitue a seven dimensional
linear subspace of ${\cal S}_8$, among many other intersting linear subspaces.
