It is known that the Hopf invariant for maps from $\mathbb{S}^{4n - 1} \to \mathbb{S}^{2n}$ is nontrivial (and captures the rational homotopy of the spheres). For $n = 1$, the Hopf fibration provides a nicely structured example of map having Hopf invariant one. The construction extends to $n = 2$ or $n=4$, and it known by a theorem of Adams that there is no map of Hopf invariant one for other values of $n$.

My question is whether there is a nice explicit construction of maps of Hopf invariant $2$ for any $n \ge 1$.