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?