Universality of Rank 6 Plücker relations and Grassmann cone preserving maps

The Plücker relations define a projective embedding of the Grassmann variety $Gr(p,n)$. We give another finite set of quadratic equations which defines the same embedding, and whose elements all have rank 6. This is achieved by constructing a certain finite set of linear maps $\bigwedge^pk^n\to\bigwedge^2k^4$, and pulling back the unique Plücker relation on $\bigwedge^2k^4$. We also give a quadratic equation depending on $(p+2)$ parameters having the same properties.