Let $V$ be a vector space and $1 \leq k \leq n$ natural numbers. 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
$\omega_{n,k} : \operatorname{Grass}_n(V) \to \operatorname{Grass}_{\begin{pmatrix}n\\k\end{pmatrix}}\left(\bigwedge^k V\right)$
as follows: We just send $V \to W$ to $\bigwedge^k V \to \bigwedge^k W$. Remark that for $k=n$ we get the usual Plücker embedding.
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 $\operatorname{\textbf{Grass}}_n(\mathcal{E}) \to \operatorname{\textbf{Grass}}_{\begin{pmatrix}n\\k\end{pmatrix}}\left(\bigwedge^k \mathcal{E}\right)$; the notation is taken from EGA I. Is this is a closed immersion (if not in general, what has to be assumed)? What else can be said about this morphism?
If we consider the morphisms $\omega_{n,k}$ all together, what structure do they impose on the familiy of functors $\operatorname{Grass}_{n}(-)$?