  Unitary transformations and density matrices are central objects in quantum
physics and various tasks require to introduce them in a parameterized form. In
the present article we present a parameterization of the unitary group
$\mathcal{U}(d)$ of arbitrary dimension $d$ which is constructed in a composite
way. We show explicitly how any element of $\mathcal{U}(d)$ can be composed of
matrix exponential functions of generalized anti-symmetric $\sigma$-matrices
and one-dimensional projectors. The specific form makes it considerably easy to
identify and discard redundant parameters in several cases. In this way,
redundancy-free density matrices of arbitrary rank $k$ can be formulated. Our
construction can also be used to derive an orthonormal basis of any
$k$-dimensional subspaces of $\mathbb{C}^d$ with the minimal number of
parameters. As an example it will be shown that this feature leads to a
significant reduction of parameters in the case of investigating distillability
of quantum states via lower bounds of an entanglement measure (the
$m$-concurrence).
