If $V$ is any finite dimensional vector field over a field $k$ and $r$ is any integer between 0 and $\dim(V)$, you can see the Grasmannian $Grass(V, r)$ as the functor $(Sch/k) \to (Sets)$ which associates to any k-scheme of finite type $S$ the set of all subsheaves $K \subset \mathcal{O}_S \otimes_k V$ such that the quotient $F:=\mathcal{O}_S \otimes_k V / K$ is locally free of rank $r$. In this way the Plucker embedding is given by

$Grass(V, r) \to \mathbb{P}(\Lambda^r V)$

$[\mathcal{O}_S \otimes_k V \to F] \to [\mathcal{O}_S \otimes_k \Lambda^rV \to \det(F)]$.

This description is slightly less intuitive than the "classical" Grasmannian (but of course it is just its relative version); on the other hand it can be naturally generalized to the case where $V$ is replaced by a coherent sheaf $\mathcal{V}$ on $S$, in particular by a vector bundle. See the book of Huybrechts and Lehn "The geometry of moduli spaces of sheaves", p. 41 for further details.

If $V$ is any finite dimensional vector space over a field $k$ and $r$ is any integer between 0 and $\dim(V)$, you can see the Grasmannian $Grass(V, r)$ as the functor $(Sch/k) \to (Sets)$ which associates to any k-scheme of finite type $S$ the set of all subsheaves $K \subset \mathcal{O}_S \otimes_k V$ such that the quotient $F:=\mathcal{O}_S \otimes_k V / K$ is locally free of rank $r$. In this way the Plucker embedding is given by

$Grass(V, r) \to \mathbb{P}(\Lambda^r V)$

$[\mathcal{O}_S \otimes_k V \to F] \to [\mathcal{O}_S \otimes_k \Lambda^rV \to \det(F)]$.

This description is slightly less intuitive than the "classical" Grasmannian (but of course it is just its relative version); on the other hand it can be naturally generalized to the case where $V$ is replaced by a coherent sheaf $\mathcal{V}$ on $S$, in particular by a vector bundle. See the book of Huybrechts and Lehn "The geometry of moduli spaces of sheaves", p. 41 for further details.