While trying to understand connections on spin manifolds, I stumbled over this (apparently obvious) statement.
Consider a spin $4$-manifold $M$. I am struggling to understand the intuition of the following statement: the Lie algebra of $\mathop{Spin}(4)$ is isomorphic to $\Lambda^2 TM$. Is there any reason as to why is this natural to consider?
Thank you for your time.