You also need a 3D vector which is orthogonal to it. An obvious second vector to use is < 0, 0, 1 > because, being on the z axis, it is already orthogonal to your e vectors which don't have any z component, and their cross product will give you a third vector which is orthogonal to both.
Incidentally, you are not setting up a MATRIX to evaluate the cross products, you are setting up a DETERMINANT. So it SHOULD be
Go from here.