I was looking for a proof of this statement
For any space $X$ the set $[X,SO(n)]$ has a natural group structure coming from the group structure in SO(n). Namely, the product of two maps $f,g:X \rightarrow SO(n)$ is the map $x \mapsto f(x)g(x)$. Similarly $[X,U(n)]$ is a group. $π_iSO(n)$ and $π_iU(n)$, special cases of the homotopy groups $π_iX$ defined for all spaces X and central to algebraic topology.
at page 27 of this book.
Are you familiar with the Eckmann-Hilton argument? In particular you want to apply it to the case that $X=S^n$, $n\geq 1$ and the two operations being $*$ given by concatenation of sphere maps, and $\cdot$ given by $(f\cdot g)(x)=f(x)g(x)$. The Eckmann-Hilton argument says that these two group operations coincide and are also abelian.