Hypermatrices, hyperdeterminants and Grassmannians.

Question

Let $Gr(k,n)$ the Grassmannian manifold of the $k$-planes in $\mathbb{C}^n$ and consider the Plucker embedding $\pi: Gr(k,n) \to \mathbb{P}(\Lambda^k \mathbb{C}^n)$. Let $A$ be the set of $n \times n$ skew-symmetric matrices and fix a natural number $s=2u$. In this way we can define the determinantal variety $Y_s:=\{\phi \in A \,|\, rank(\phi) \le s\}$. It is very easy to verify that there cone over $Gr(2,n)$, say $\Gamma(Gr(2,n))$ is isomorphic to the determinantal variety defined by the vanishing of $4 \times 4$ pfaffians. It is also well know that the ideal defining Grassmannian is given by quadratic relations: the Plucker relations, so I think that it is impossible to give an isomorphism between $\Gamma(Gr(k,n))$ and $Y_s$ is $k \ne2$. Now, my questions are: 1) Is it possible to give a notion of skew-symmetric hypermatrix? 2) Is it possible to give a notion of (skew-symmetric) hyperdeterminantal variety? (In some sense...) 3) Is it possible to realize the cone over Grassmannian ($\Gamma(Gr(k,n))$) as a hyperdeterminantal variety? My question is motivated by the fact that, roughly speaking, I need more indexes to fit into a matrix the Plucker relations. I'm sorry if my question is not very precise, thank you for the help!