  We show that decoding of $\ell$-Interleaved Gabidulin codes, as well as
list-$\ell$ decoding of Mahdavifar--Vardy codes can be performed by row
reducing skew polynomial matrices. Inspired by row reduction of $\F[x]$
matrices, we develop a general and flexible approach of transforming matrices
over skew polynomial rings into a certain reduced form. We apply this to solve
generalised shift register problems over skew polynomial rings which occur in
decoding $\ell$-Interleaved Gabidulin codes. We obtain an algorithm with
complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem
and is proportional to the code length $n$ in the case of decoding. Further, we
show how to perform the interpolation step of list-$\ell$-decoding
Mahdavifar--Vardy codes in complexity $O(\ell n^2)$, where $n$ is the number of
interpolation constraints.
