The problem some rotation representations run into
One common alternative to a rotation matrix is describing orientation as three separate rotation angles, one around each axis, often called Euler angles. They're intuitive to read, but they carry a real hazard called gimbal lock. Gimbal lock happens when two of the three rotation axes swing around to line up with each other, at which point rotating around either of them produces the same effect, and a degree of freedom that should be independent is effectively lost.
This isn't a hypothetical edge case, it happens at specific, reachable orientations, and a robot approaching one can suddenly find that a motion it needs to make becomes numerically unstable or undefined using that representation. Historically, this exact issue affected inertial guidance systems, and it remains a real practical concern for any orientation representation built out of a sequence of separate axis rotations.