Maybe it is some kind of stupid idea and I have not thought too much about it, but would it not be possible to construct something like the determinant of your matrix (with vectorial entries) with sums (with sign) of all possible combinations of vectorial products? Anycase, not sure if this would help because the transformations that preserve determinant do not break rows or columns (and you needed to mix the basis).
Anyway, I find it very curious and interesting.

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