One can verify that this formula is equivalent to the known formulas for the two- and three-dimensional case by using the following facts: In three dimensions, ω ω ( u , v , w ) = g ( u × × v , w ) and in two dimensions, ω ω ( v , w ) = g ( R v , w ) , where R ∈ ∈ SO ( 2 ) ⊂ ⊂ L ( V , V ) is the rotation by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation).