A question on Grassmann functor

For a commutative ring $R$ and an $R$-module $M$, let $\operatorname{Gr}(n,M)$ be the Grassmann functor from the category of $R$-algebras to sets.

As I understand it, it should be possible to define a subfunctor $U$ when fixing $x_1, \ldots, x_n$ elements of $M$, how does this subfunctor look like? Is it an open subfunctor of the Grassmann functor that yields an open cover of $\operatorname{Gr}(n,M)$?

Sorry for my vaguely asked question, I would appreciate any reference where something similar to the above is done.