A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid has 10 nodes and the family of such surfaces seem to have the same dimension as the moduli space of 10 points in $\mathbb{P}^3$.

QUESTION: is any general set of 10 points in $\mathbb{P}^3$ the singular locus of a symmetroid? Or better, how can one detect whether such a point set comes from a symmetroid?