Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Question

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it possible to prove that $C$ is an open set of a determinantal variety? If yes, how can I prove it? Thank you!