Note that x ↦ a × x is linear, so it is determined by its action on basis elements. But by direct calculation ( A − − A T ) e 1 = [ 0 a 3 − − a 2 ] = a × × e 1 ( A − − A T ) e 2 = [ − − a 3 0 a 1 ] = a × × e 2 ( A − − A T ) e 3 = [ a 2 − − a 1 0 ] = a × × e 3 Here, { e 1, e 2, e 3 } represents an orthonormal basis in the coordinate directions of R 3 .