  We present three different matrix bases that can be used to decompose density
matrices of $d$--dimensional quantum systems, so-called qudits: the
\emph{generalized Gell-Mann matrix basis}, the \emph{polarization operator
basis}, and the \emph{Weyl operator basis}. Such a decomposition can be
identified with a vector --the Bloch vector, i.e. a generalization of the well
known qubit case-- and is a convenient expression for comparison with
measurable quantities and for explicit calculations avoiding the handling of
large matrices. We present a new method to decompose density matrices via
so--called standard matrices, consider the important case of an isotropic
two--qudit state and decompose it according to each basis. In case of qutrits
we show a representation of an entanglement witness in terms of expectation
values of spin 1 measurements, which is appropriate for an experimental
realization.
