The formula a \wedge b = (ab-ba)/2 only applies if a and b are vectors by the way, it’s not a general formula. The most general form of that expression is more complicated and involves a sum of more terms for a grade 3 element, and the formula simply does not apply to grade 3 and higher products.

You can find the most general form of the expression if you study my notes at Algebra · Grassmann.jl

Doran and a lot of other people fail to tell people that these formulas only apply in limited context, and are not the most general expression.