By Serre's theorem, we know the only nontorsion parts of the homotopy groups of spheres occur as $\pi_n(S^n)$ and $\pi_{4n-1}(S^{2n})$. The first of these are trivial to describe, but the second have very interesting, symmetric incarnations, they are the generalised hopf fibrations, at least for $n=1,2,4$, associated to the real normed division algebras.

Are there a similar explicit descriptions for representatives of these higher nontorsion elements too?

Even if we don't have explicit descriptions, do we know anything about the values of the hopf invariants associated to these maps?