  This paper extends the study of rank-metric codes in extension fields
$\mathbb{L}$ equipped with an arbitrary Galois group $G =
\mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these
codes as subspaces of the group algebra $\mathbb{L}[G]$, and we relate this
point of view with usual notions of rank-metric codes in $\mathbb{L}^N$ or in
$\mathbb{K}^{N\times N}$, where $N = [\mathbb{L} : \mathbb{K}]$. We then adapt
the notion of error-correcting pairs to this context, in order to provide a
non-trivial decoding algorithm for these codes. We then focus on the case where
$G$ is abelian, which leads us to see codewords as elements of a multivariate
skew polynomial ring. We prove that we can bound the dimension of the vector
space of zeroes of these polynomials, depending of their degree. This result
can be seen as an analogue of Alon-F\"uredi theorem -- and by means, of
Schwartz-Zippel lemma -- in the rank metric. Finally, we construct the
counterparts of Reed-Muller codes in the rank metric, and we give their
parameters. We also show the connection between these codes and classical
Reed-Muller codes in the case where $\mathbb{L}$ is a Kummer extension.
