The candidates include:
Looking more closely, the fourth representation gives parameters in R3. The second gives parameters in S2×S1; if we replace the unit vector by the actual axis of rotation, so that n and -n give the same axis line, this becomes RP2×S1, where RP2 is the real projective plane.
That makes four or five manifolds that are used to try to give charts on SO(3). The truth about it, so to speak, is that it is diffeomorphic to RP3: the quaternion representation is precisely a two-to-one mapping from S3 to SO(3). This suggests that it has certain theoretical advantages; and also that conversions from other representations to it will encounter chart problems.
One area in which these considerations, in some form, become inevitable, is the kinematics of a rigid body. One can take as definition the idea of a curve in the Euclidean group E(3) of three-dimensional Euclidean space, starting at the identity (initial position). The translation subgroup T of E(3) is a normal subgroup, with quotient SO(3) if we look at the subgroup E+(3) of direct isometries only (which is reasonable in kinematics). Therefore any rigid body movement leads directly to SO(3), when we factor out the translational part.