Polyvector representations of $\operatorname{GL}_n$

In the present paper we characterise $\bigwedge^n(\operatorname{GL}(n,R))$ over any commutative ring $R$ as the connected component of the stabiliser of Plücker ideal. This folk theorem is classically known for algebraically closed fields and should be also well-known in general. However, we are not aware of any obvious reference, so we produce a detailed proof which follows a general scheme developed by W. C. Waterhouse. The present paper is a technical preliminary for a subsequent paper, where we construct decomposition of transvections in polyvector representations of $\operatorname{GL}_n$.