What do you mean by "real life"?
You can think of the vector cross product as a determinant. The vector cross product shows up in electricity and magnetism, in Maxwell's Equations, as well as other places.
As a somewhat less spectacular example, you can think of Kramer's Rule for solving linear systems of equations. Kramer's Rule has more theoretical interest than practical, because computing determinants using, say, a Jacobian expansion is computationally intensive. For large problems, you'd probably try to diagonalize, or at least upper-triangularize (if that's a word) a matrix in order to compute its determinant as the product of the elements along the main diagonal.