I have found that calculating the determinant of a 3x3 matrix with vector operations is easier to remember, easier to express and easier to code than the usual way it is learned.

Given a 3x3 matrix:

We construct the following three vectors, one for each row of the matrix:

The determinant of the matrix can be calculated with any of the following formulas:

If we take one vector for each column of the matrix, the same formulas apply. The expression $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ is called scalar triple product.

An implementation example in pseudo-code: