The orientation of the wedge product of two vectors has two possibilities (positive or negative). We sometimes imagine that this corresponds to the “handedness.” When we generalize this to higher dimensional wedges, the orientation still has two possibilities.
Is there any way to have a generalization of orientation which has more than two possibilities? That is, $w \wedge u$ can be positive or negative, with $u \wedge w$ having the opposite orientation. But might $u \wedge w \wedge \ y$ have three possible orientations? Or perhaps $3!$ or $3!/[(3-1)! 1!]$ signed orientations? Of course, this would no longer mean the same thing as the previous definition of orientation, but it intuitively seems like there might be multiple ways to “orient” higher dimensional objects...
