Spin group without Clifford algebras

Question

I have to build the spin group $Spin(n)$ without use Clifford algebras. Can I find a complete description of spin group with a topological method? How can I build $Spin(n)$ as the double covering of $SO(n)$?

Answer 1

Define $Spin(n)$ as the universal cover of $SO(n)$. The universal cover of a Lie group is a Lie group (e.g. Theorem 2.5 on p. 10 of these notes). In particular, when $n\geq 3$, the homotopy exact sequence gives $\pi_1(SO(n)) = \mathbb{Z}_2$, so $Spin(n)$ is a double covering of $SO(n)$.