You can let the matrix act by ordinary matrix multiplication on ordinary vectors in three-dimensional space.
This will transform a vector in a triple containing the original vector and the lengths of the two projections on $a$ and $b$.
While I feel that this counts as "any significance", it isn't very satisfactory, because the matrix, as you presented it, does not allow for matrix multiplication.
I think it is a much more useful point of view, to first view $i$,$j$,$k$ as three scalar variables (better denoted by $x$, $y$, $z$), then take the determinant of your matrix and then regard the cross product as the gradient vector this determinant.