Is the Cone over Grassmannian manifold a determinantal variety?

Question

Let consider the Grassmann manifold $Gr(k,n)$ in the Plucker embedding and the Cone over $Gr(k,n)$, say $C(Gr(k,n))$. On the other hand consider $M$ the set of $n \times n$ skew-symmetric matrices. Take $s=2u$ with $u \in \mathbb{N} \cup \{0\}$ and consider the determinantal variety $Y_s=\{\phi \in M \,|\,rank(\phi)\le s\}$. Is $C(Gr(k,n))$ isomorphic to $Y_s$ for some $(k,n,s)$? Thank you!