"In mathematics the spin group Spin(n) 1 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, n (n − 1)/2, and its Lie algebra with the special orthogonal group. For n > 2 , Spin(n) is simply connected and so coincides with the universal cover of SO(n)." taken from wikipedia
How to prove that $ Spin(n) $ exists for any $ n $ .