  We consider the Grassman manifold $G(E)$ as the subset of all orthogonal
projections of a given Euclidean space $E$ and obtain some explicit formulas
concerning the differential geometry of $G(E)$ as a submanifold of $L(E,E)$
endowed with the Hilbert-Schmidt inner product. Most of these formulas can be
naturally extended to the infinite dimensional Hilbert space case.
