> You can do rotation with a 3x3 matrix.
You can do a rotation or some rotations but SO(3) is not simply connected.
It mostly works for rigid bodies centered on the origin, but gimbal lock or Dirac's Plate Trick are good counter example lenses. Twirling a baton or a lasso will show that 720 degrees is the invariant rotation in SO(3)
The point at infinity with a 4x4 matrix is one solution, SU(3), quaternions, or recently geometric product are other options with benefits at the cost of complexity.
I think you are confused about what 'simply connected' means. A 3x3 matrix can represent any rotation. Also from a given rotation there is a path through the space of rotations to any other rotation. It's just that some paths can't be smoothly mapped to some other paths.
SO(3) contains all of the orthogonal 3x3 matrices of determinant 1.
If you are dealing with rigid bodies rotated though the origin like with the product of linear translations you can avoid the problem. At least with an orthonormal basis R^3 with an orthogonal real valued 3x3 matrix real entries which, where the product of it with its transpose produces the identity matrix and with determinant 1
But as soon as you are dealing with balls, where the magnitude can be from the origin to the radius, you run into the issue that the antipodes are actually the same point, consider the north and south poles being the same point, that is what I am saying when the topology is not simply connected.
The rigid body rotation about the origin is just a special case.
Twist a belt twice and tape one end to the table and you can untwist it with just horizontal translation, twist it once (360deg) and you cannot.