There is a very natural way to define generators of $\pi_{4n-1}(SO(4n))\cong \mathbb{Z}\oplus \mathbb{Z}$ in terms of quaternions when $n=1$ and octonions when $n=2$ (see for example Tamura, *On Pontrjagin classes and homotopy types of manifolds*, 1957). Since there are no normed division algebras in higher dimensions, it is not possible to do the same for $n>2$.

I was wondering whether there still is a natural identification between $\pi_{4n-1}(SO(4n))$ and $\mathbb{Z}\oplus\mathbb{Z}$ using explicit generators when $n>2$?