How to solve the matrix from known data?

If $A$ is a $n \times n$ nonsingular matrix, and $\det \left( {\begin{array}{*{20}{c}} {{a_{ij}}} & {{a_{ik}}} \\ {{a_{lj}}} & {{a_{lk}}} \\ \end{array}} \right) = {c_{ijkl}}$ for all $i,j,k,l \in \{ 1,2, \cdots n\}$, then can we solve all $a_{ij}$ in terms of $c_{ijkl}$?

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Naively: we have a system of homogeneous $n^2$-variate polynomials of the form: $x_i x_j - x_k x_l = c_m.$ And then Groebner basis.. –  user2468 Mar 16 '12 at 4:34
Presumably $c_{ijkl} = -c_{ljki} = -c_{ikjl}$. –  Robert Israel Mar 16 '12 at 7:34

For large $n$ it's a highly overdetermined system, with ${n \choose 2}^2$ equations in $n^2$ unknowns. For $n=2$, on the other hand, there's only one nontrivial equation, which just gives you the determinant of $A$. And of course you can multiply the whole matrix by $-1$ without changing any of those determinants.