Podles defined quantum 2-spheres in such a universal way. In a special case they restrict to $C(S^2)$. The description in this case is: the universal unital C*-algebra generated by operators $A$ and $B$ satisyfing
- $A^* = A$
- $AB = BA$
- $BB^* = B^*B = I - A^2$.
The generalization to higher dimensions can be found in Section 2 of this paper.
In the case of $\mathbb{R}^n$, remember that $\mathbb{R}^n$ is homeomorphic to $S^n$ minus a point. So $C_0(\mathbb{R}^n)$ is realized as (any) maximal ideal of $C(S^n)$.