We prove that the norm of the Euler class for flat vector bundles is (in even dimension , 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 and taking only the two values ; a null-set obstruction prevents any cocycle from existing on the projective space. We establish the uniqueness of an antisymmetric representative for in bounded cohomology.