I think angular velocities are precisely the bivectors (which I take to be the grade-2 elements of an exterior algebra or Geometric Algebra). The change of basi

I think angular velocities are precisely the bivectors (which I take to be the grade-2 elements of an exterior algebra or Geometric Algebra). The change of basis is indeed different than for ordinary vectors. The exponential map (the obvious generalisation of e^x) then takes bivectors to the "rotors", which represent rotation operators. In terms of Lie theory, the rotors are a Lie group and the bivectors are the corresponding Lie algebra.