Unit sphere fibrations in Euclidean space

We show that if an open set in ${\mathbb{R}}^d$ can be fibered by unit $n$-spheres, then $d\ge 2n+1$, and if $d=2n+1$, then the spheres must be pairwise linked, and $n\in \{0,1,3,7\}$. For these values of $n$, we construct unit $n$-sphere fibrations in ${\mathbb{R}}^{2n+1}$.