How do we generalize the cross product to other dimensions?

I should maybe have notated the canonical vectors in a way so we could infer their dimensions from their writings :P I have a math blog if you are interested: matekon.tumblr.com The opening and closing jingles are my compositions. You can hear more of them on my soundcloud: soundcloud.com/matekon This video was made with a Zoom H2n recorder and the softwares Shotcut, Reaper and Powerpoint. The graphics are made with LaTeX.