I would like to get an explicit expression for the determinant of a Jacobian $$J_{ij} = \delta_{ij}\left(\sigma^{2}+\sum_{k}w_{ik}y_{k}^{2}\right)-w_{ij}y_{i}y_{j},$$ where $i,j = 1,...,n$, $w_{ij}\ge 0$ and $\delta_{ij}$ is the Kronecker delta.

I was able to find explicit expression for special cases of $w_{ij}$ but not for the general case.

Any hint to a solution would be greatly appreciated.