The set of jumping conics of a locally free sheaf of dimension 2 on P²

We consider a locally free sheaf $F$ of dimension 2 on $P^2$. A conic $q$ on $P^2$ is called a jumping conic if the restriction of $F$ to $q$ is not the generic one. We prove that the set of jumping conics is the maximal determinantal variety of a skew form. Some properties of this skew form are found.