>>11242250>>11242331I'm not saying that this would be a good or even the best definition, nor do I want to imply to take Wikipedia as your primary source of definition.
But SO(3) is defined as the isometric transformations, which just as well could be seen as a set of functions which as domain have all the points in 3D space (or even objects too) and as outputs have the origin-rotated versions of that, ,
The fact that it suffices to specify 3 point coordinates (a choice or orthonormal vectors, and not all points, let alone not extended objects such as cubes) to characterize a rotation
doesn't have to be assumed to say that SO(3) are all rotations.
You can define G=SO(3) just as the group of transformations and their induced concatenation as their group product, without any reference to matrix multiplication and bases etc. upfront. A map from this group G into the set of all orthogonal matrices can still be defined.
