Let $A=(a_{ij})$ be an invertible matrix. I would like to study the rank of the following matrix $$M=\Big(a_{ik}a_{jk}\Big)_{1\leq i<j\leq n, 1\leq k\leq n}.$$ For $n=3$ I used the page Matrix Calculator to compute the determinant of $M=\begin{pmatrix}a_{11}a_{21}&a_{11}a_{31}&a_{21}a_{31}\\ a_{12}a_{22}&a_{12}a_{32}&a_{22}a_{32}\\ a_{13}a_{23}&a_{13}a_{33}&a_{23}a_{33}\end{pmatrix}$ and I get a long formula $$\det M=a_{33}a_{11}a_{12}a_{21}a_{23}a_{32}-a_{12}a_{13}a_{21}a_{23}a_{31}a_{32}+a_{33}a_{12}a_{13}a_{21}a_{31}a_{22}-a_{33}a_{11}a_{12}a_{23}a_{31}a_{22}-\ldots+\ldots$$
Is it possible to write the determinants $n\times n$, extracted from $M$, using some quadratic forms?
It's related to my question in Mathoverflow Degenerate representation