"In mathematics the spin group $\text{Spin}(n)$ is the double cover of the special orthogonal group $SO(n)$, such that there exists a short exact sequence of Lie groups...
As a Lie group Spin(n) therefore shares its dimension, $\frac{n(n − 1)}2$, and its Lie algebra with the special orthogonal group. For $n > 2$ , $\text{Spin}(n)$ is simply connected and so coincides with the universal cover of $SO(n)$." (taken from wikipedia)
How do I prove that $\text{Spin}(n) $ exists for any $ n $?