I'm studying special relativity for the second time and there's something I think I didn't get since I studied classical mechanics: the idea of a frame of reference. I think the underlying idea is that of a point of view, so that we want to study some phenomenon, but we have to observe it so that a natural question should be: we are observing it from which point of view? That "point of view" should be a frame of reference. The point, however, is that I don't understand how to transition this to mathematics. Many books seems to correspond frames of references and coordinate systems, but in the same moment they refer as a collection of axes. But if we are explicitly talking about axes, we are assuming that the coordinate system is cartesian. On the other hand, we usually talk about the motion of a reference frame, so that it would mean the movement of a coordinate system. All of this would work well in euclidean three-space. In that manifold, there's a natural identification between points and coordinates. Indeed if we move from point to point, we could still draw axes on the other point and get the same system and the notion of movement of such system. But on other manifolds, it doesn't work well, because there would be no natural notion of moving a coordinate system. So, my questions are: is my intuitive idea of a reference frame correct? What, mathematically, really is a reference frame? And I'm asking those questions in a general setting, so that I'm also thinking about classical mechanics (with no idea of spacetime).