  In this paper, we construct and study a Serre-type spectral sequence for
motivic cohomology associated to a map of bisimplicial schemes with motivically
cellular fiber. Then, we show how to apply it in order to approach the
computation of the motivic cohomology of the Nisnevich classifying space of
projective general linear groups. This naturally yields an explicit description
of the motive of a Severi-Brauer variety in terms of twisted motives of its
\v{C}ech simplicial scheme.
