Let $V$ be any vector space and $1 \leq k \leq n$. By $\operatorname{Grass}_n(V)$ I mean the Grassmannian of $n$-codimensional subspaces of $V$, that is, $n$-dimensional quotients of $V$. Then we can define a natural map
$$\textstyle \omega_{n,k} : \operatorname{Grass}_n(V) \to \operatorname{Grass}_{\Large\binom{n}{k}}\left(\bigwedge^k V\right), \quad (q : V \twoheadrightarrow W) \mapsto (\bigwedge^k q : \bigwedge^k V \twoheadrightarrow \bigwedge^k W).$$
Remark that for $k=n$ we get the usual Plücker embedding $\mathrm{Grass}_n(V) \hookrightarrow \mathbb{P}(\Lambda^n V)$.
Questions. How are the maps $\omega_{n,k}$ called? Are they also embeddings? Can you give me some reference where they are studied?
More generally, for a quasi-coherent module $\mathcal{E}$ on a scheme $S$ we get a morphism of $S$-schemes
$$\textstyle \omega_{n,k} : \operatorname{\textbf{Grass}}_n(\mathcal{E}) \to \operatorname{\textbf{Grass}}_{\Large \binom{n}{k}}\left(\bigwedge^k \mathcal{E}\right).$$ The notation is taken from EGA I, §9.7. Is this is a closed immersion? If not, is it the case under some additional assumption? What else can be said about this morphism?