I am trying to understand Atiyah's paper "Algebraic Topology and Elliptic Operators" (link is pay-walled). In the paper he defines the degree of a map from $S^{2n-1}\rightarrow GL(n,\mathbb{C})$ by mapping to the first column and then normalizing to obtain a map $S^{2n-1}\rightarrow S^{2n-1}$ which has the usual notion of degree.
He claims that the degree of such a map will be a multiple of $(n-1)!$. Can anyone explain why this should be true? I am also interested in a reference with this notion of degree.