How to understand that affine cone of Grassmannian?

Question

I am reading the paper, On page 5, it is said that the affine cone of a Grassmannian $Gr(k,n)$ is the affine subvariety of decomposable $k$-forms in $\Lambda^k \mathbb{C}^n$. I am trying to understand this. By definition, the affine cone of $Gr(k,n)$ is $\{0\} \cup \pi^{-1}(Gr(k,n))$, where $\pi$ is the canonical projection from the affine space whose dimension is greater than $Gr(k,n)$ by 1 to $Gr(k,n)$. Why the affine cone of $Gr(k,n)$ is the affine subvariety of decomposable $k$-forms in $\Lambda^k \mathbb{C}^n$? Thank you very much.