One general way to derive this formula is to consider the signed area of the parallelogram built on vectors
where , are the vectors of an orthonormal positively oriented basis. The signed area function takes two vectors and and returns a number. It has the following properties.
(i.e., the function is linear in both arguments)
If you expand (*) according to linearity, you should get the required formula.
For more, see the shoelace formula in Wikipedia.