$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns.

Consider the following $\frac{n(n+1)}{2}$ overdetermined bilinear equations:

$\sum_{\substack{j=1,}{j \ne i}}^{n} x_{i}y_{j} = n_{ii} + x_{i}y_{i} \in \mathbb{Z}$.

$x_{i}y_{j} - x_{j}y_{i} = n_{ij} \in \mathbb{Z}$ when $i < j$.

When is the system solvable and when is it solvable over $\mathbb{Z}$?