The properties of the exterior algebra $\Lambda(\mathbb R^n)$ studied in the paper are related to the Euclidean structure in this algebra induced by the scalar product in $\mathbb R^n$. A geometric interpretation of the interior multiplication for decomposable polyvectors is given. The Cartan criterion of decomposability for the polyvectors is formulated in a coordinateless form. The Pluccer model of the real Grassmannian manifold is realized as a submanifold of the Euclidean space $\Lambda(\mathbb R^n)$, and the isometry of this submanifold onto the classical Grassmannian manifold with $SO(n)$-invariant metric is indicated. For the bivectors the canonical decomposition is described.