One often reads about oriented volumes as a motivation for determinants. In two dimensions you can scetch some nice pictures which may convince students that it is a good idea to have a closer look at alternating multi-linear functionals.

However, my feeling is that you are cheating because, if you come to the point where you really define orientations of vector spaces you use determinants.

Hence the question: Is there a definition of orinetation which does not rely one determinants?