Quadrics defined by skew-symmetric matrices

In this paper we propose a model for computing a minimal free resolution for ideals of the form $I_1(X_n{Y}_n)$, where $X_n$ is an $n\times n$ skew-symmetric matrix with indeterminate entries $x_{ij}$ and $Y_n$ is a generic column matrix with indeterminate entries $y_j$. We verify that the model works for $n=3$ and $n=4$ and pose some statements as conjectures. Answering the conjectures in affirmative would enable us to compute a minimal free resolution for general $n$.