We study the real Grassmannian $G^\mathbb R _{n,n}$ of $n$-planes in $\mathbb R ^{2n}$, with $n\ge 3$, and its reduced space. The latter is the irreducible symmetric space $\bar G^\mathbb R _{n,n}$, which is the quotient of the space $G^\mathbb R _{n,n}$ under the action of its isometry which sends a $n$-plane into its orthogonal complement. One of the main results of this monograph asserts that the irreducible symmetric space $\bar G^\mathbb R _{3,3}$ possesses non-trivial infinitesimal isospectral deformations ; it provides us with the first example of an irreducible reduced symmetric space which admits such deformations. We also give a criterion for the exactness of a form of degree one on $\bar G^\mathbb R _{n,n}$ in terms of a Radon transform.