Self-dual two-forms in \(2n\)-dimensional space satisfying the Yang-Mills equations determine a \((n^2-n+1)\)-dimensional manifold \({\mathcal S}_{2n}\) (of antisymmetric matrices such that \(A^2+\lambda^2I=0)\). The dimension of maximal linear subspaces of \({\mathcal S}_{2n}\) is equal to the Radon-Hurwitz number of linearly independent vector fields on the sphere \({\mathcal S}^{2n-1}\). One exhibits \((2^c-1)\)-dimensional subspaces in dimensions which are multiples of \(2^c\), for \(c=1, 2, 3\). It is remarkable that the seven-dimensional linear subspaces of \({\mathcal S}_8\) include the self-dual two-forms of E. Corrigan, C. Devchand, D. B. Fairlie and J. Nuyts [Nuclear Phys. B 214, 452-464 (1983; MR 84i:81058)]. The relation of the linear subspaces with the representation of Clifford algebras and octonionic instantons is discussed (see also [D. B. Fairlie and J. Nuyts, J. Phys. A 17, 2867-2872 (1984; MR 86e:53056)] and [S. Fubini and H. Nicolai, Phys. Lett. B 155, 369-372 (1985; MR 86m:81101)].