Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?

Yes, it can be defined as the univeral commutative $C^*$-algebra with unit, generated by $n+1$ self adjoint elements $x_1,...,x_{n+1}$ subject to the relation $x_1^2+...+x_{n+1}^2=1$. Here universal mean the following: $A$ with generators $(a_j)_j$ and relations $(r_k)_k$ is called universal if whenever there is $C^*$-algebra $B$ generated by $(b_j)_j$ (the same indexing set) and satysfying relations $(r_k)_k$ then there is an epimorphim $\varphi:A \to B$ such that $\varphi(a_j)=b_j$. The problem with existence of such universal algebras is because some relations don't impose any restriction on norms of elements. For example there is no universal unital $C^*$-algebra generated by a single self adjoint element but there is universal unital $C^*$-algebra generated by a single unitary element (this algebra is in fact $C(S^1)$). However you can prove that if there is any $C^*$-algebra $B$ with generating set $S=(b_s)_s$ such that $\|b_s\| \leq 1$ and those elements satisfy some relations $(r_j)_j$ then there is a universal $C^*$-algebra $A$ with generating set $(a_s)_s$ (the same cardinality as $S$) where $(a_s)_s$ satisfy relations $r_j$. So this difficulty concerning the norms is somehow the only obstruction

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 $B$ satisfying

• $A^* = A$
• $AB = BA$
• $BB^* = B^*B = I - A^2$.

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 $C(S^n)$.