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 is called scalar triple product.
An implementation example in pseudo-code: