I'm looking for a proof of the fact that if $V$ is a finitely dimensional vector space, then $G_p(V) \setminus \{0\}$ is a submanifold of $\Lambda_pV$.
Here $G_p(V) = \{ v_1 \wedge ... \wedge v_p \ | \ v_1, ..., v_p \in V \}$ for $p \ge 1$ and $G_0(V) = \mathbb{K}$.
I know that Grassmannian is a manifold but I don't know how to prove that it is a submanifold of $\Lambda_pV$.
I've found that it can be embedded in the projectivization $\mathbb{P}(\Lambda _k V)$ of $\Lambda _k V$ .
Could this be what I'm after?
Thank you for your help.