But couldn't one use two unit quaternions to describe rotations in 4d space? An antisymmetric matrix in 4 dimensions has N(N-1)/2 = 6 independent variables. But

But couldn't one use two unit quaternions to describe rotations in 4d space? An antisymmetric matrix in 4 dimensions has N(N-1)/2 = 6 independent variables. But each unit quaternion has three independent variables, so two of them would be enough to describe rotations in 4d.