You probably know what i am going to say, but here goes
The term dimensions in space , geometric dimensions etc are all buzzwords, created by our scientific past, and our science fiction past. Ihave no objection to that as long as one does not confuse that with what we can actually perceive.
The real analysis of space belongs to Grassmann, and he says quite tellingly that we are not restricted to 3 dimensions. We live in an n-dimensional space, and we can describe an n-dimensional space if we want to. Grassmann wanted to and he did.
The solution you seek is that of Lagrange, and it is usually called generalised coordinates. It is of course easier to represent generalised coordinated in our liin space than on a flat surface, a projection from our living space.
The book Adventures in Flatland has a lot to answer for in perpetuating this mythology! The orthogonal set of axes is in fact a set of 6 axes, and five rotational dimensions, yes dimensions, and one of the axes has to be defined as the generator of fundamental dimension of the set, in order to establish a fundamental orientation. In addition we have to establish a common formalism to avoid confusion even in this familiar framework. This reference frame is Cartesian because it developed along the lines of Descartes proposal. Nowadays most mathematical physicists work free of the Cartesian system exactly because it is such a Special system. It has 11 or 12 dimensions depending on how you want to count. 3d is and always has been a synonym for our living space, not a description of its dimensional structure.
There is another use of the word dimension in dimensional analysis. These dimensions are standard quantities of measure and relate back to Grassmann through Riemann and Gauss. The SI syatem was set up to establish a common system of measures, and tese were recognised as dimensions precisely because the dimension, that is cut or quantize space.
Quantizing space has to be understood too as dealing with our experience of it in terms of the notion of extension. Extension was the Cartesian notion that leavin god out of the picture, our reality is basically objects and things extending themselves everywhere! Of course this is a simple explanation of what he meant, but nevertheless it generated considerable philosophical debate, only really answered by Newton's Principles of Mathematics, in the sense thar everyone eventually went along with Newton. Leibniz had developed a theory of Monads, but it never really caught on. Extension had to be quantified, and consequently dimensions were created from extension, philosophically.
Of course, any artisan knows this innately, it is just certain thinkers get an idea that goes against our common experience and then we spend literally centuries straightening it out!