I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps $S^n \rightarrow S^n$ of degree $k$, we will then be able to get explicit representatives of all elements of the group. I suspect that one way to do this would be to follow Bott's proof of periodicity via minimal geodesics on $SU(n)$ and see what it gives you but I haven't been able to complete this idea.
Alternatively, what is a representative of $\pi_n(U)$? Maybe it is easier to describe corresponding vector bundle over $S^{n+1}$ (using the clutching construction)?
Side remark: what I really need is a function on the unit $n$-disk $f: D^n \rightarrow \mathbb{C}^n$ such that $df(x) \in U(n)$ for $x\in D^n, df(x) = Id$ for $x\in \partial D^n$. Then $df$ defines an element of $\pi_n(U(n))\cong \mathbb{Z}$ and I want $df$ to represent the generator. It might be easier to explicitly construct such an $f$ than directly constructing the map $S^n \rightarrow U(n)$.