About embeddings of real grassmannians

Question

I am dealing with the embedding of real grassmannians $G(n,k)$ on $\mathbb{R}^{n^2}$ via the map associating to each vector space the projection matrix on it in the canonical base of $\mathbb{R}^n$. More explicitely this map is defined identifying: \begin{equation} G(n,k)=\{A\in\mathbb{R}^{n^2}\,|\,A^2=A,\,A^{T}=A,\, trace(A)=k\}. \end{equation} In ''Real Algebraic Geometry'' by Bochnak, Coste, Roy they prove that the map defined as before is a birational isomorphism between the grassmannian, as an abstract variety, and the algebraic set defined above. Is it also true that, as for the Plucker embedding, the equations defining this algebraic set are also generators for the vanishing ideal of polynomials? Is there a reference where to find something about it? Are there other ways to embed grassmannians getting that the equations defining the affine algebraic set are also generators of the vanishing ideal of polynomials?