  Let G(k,n) be the Grassmannian of oriented subspaces of dimension k of R^n
with its canonical Riemannian metric. We study the energy of maps assigning to
each P \in G(k,n) a unit vector normal to P. They are sections of a sphere
bundle E_{k,n}^1 over G(k,n). The octonionic double and triple cross products
induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively.
We prove that they are harmonic maps into E_{k,n}^1 endowed with the Sasaki
metric. This, together with the well-known result that Hopf vector fields on
odd dimensional spheres are harmonic maps into their unit tangent bundles,
allows us to conclude that all unit normal sections of the Grassmannians
associated with cross products are harmonic. In a second instance we analyze
the energy of maps assigning an orthogonal complex structure J(P) on P^{\bot}
to each P\in G(2,8). We prove that the one induced by the octonionic triple
product is a harmonic map into a suitable sphere bundle over G(2,8). This
generalizes the harmonicity of the canonical almost complex structure of S^6.
