The geometry of the sphere of the gauges of degree two

We construct a canonical bijection between the set of extremal points of the unit sphere $S^*_{2,2k}\subset\Lambda^2(\mathbb R^{2k})$ (in the norm of cocharge) and the manifold of orthogonal comples structures in $\mathbb R^{2k}$. Under this bijection the unitary basises correspond to the expansion of the forms which $re$ the conjugate norm of the charge. Using this correspondence we give the classification of $t$ facets of the sphere $S^*_{2,n}$ and the known classification of the facets of the set which is polar to that sphere.