  For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the
conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the
complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of
skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to
describe a natural stratification of the twistor space
$\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have
called {\it generalised complex orthogonal Stiefel manifolds} of
$\mathbb{C}^{n+1}$. In particular, the twistor space
$\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic to a non-singular complex
quadric hypersurface in $\mathbb{P}^{7}$. We explicitly construct a
real-analytic foliation, by linear 3-folds, of this quadric hypersurface such
that the quotient space is isomorphic to the 6-sphere with its standard metric.
This foliation is Riemannian with respect to the Fubini-Study metric and
isometrically equivalent to the twistor fibration over the 6-sphere.
