Embeddings of Grassmannians realized by identification of a $k$-dimensional subspace of $\Bbb F^m$ with the orthogonal projection onto it are studied. It is shown that such an embedding has parallel second fundamental form and embeds the Grassmannian minimally into a hypersphere of certain Euclidean space of matrices. A result on holomorphic sectional curvature of the complex Grassmannian is also proved.