I am looking for a non-permutational definition of determinant. The definition should have these properties:

1: Calculational power (easily applicable, it cold be used for practical calculations).

2: It should hold the usual properties (otherwise it wouldn't be a determinant, right!)

3: No permutation, No permutation, please no permutations.

I would also appreciate if you apply the definition and calculate the determinant of a $4×4$ matrix.

For all I care the definition could be from an obsolete parchment, but it needs to have those three properties.

Appreciate all the help!