  We prove that the norm of the Euler class E for flat vector bundles is
$2^{-n}$ (in even dimension $n$, since it vanishes in odd dimension). This
shows that the Sullivan--Smillie bound considered by Gromov and Ivanov--Turaev
is sharp. We construct a new cocycle representing E and taking only the two
values $\pm 2^{-n}$; a null-set obstruction prevents any cocycle from existing
on the projective space. We establish the uniqueness of an antisymmetric
representative for E in bounded cohomology.
