  Electromagnetic media with generic linear response provide a rich class of
Lorentz violation models. In the framework of a general covariant metric-free
approach, we study electromagnetic wave propagation in these media. We define
the notion of an optic tensor and present its unique canonical irreducible
decomposition into the principle and skewon parts. The skewon contribution to
the Minkowski vacuum is a subject that does not arise in the ordinary models of
Lorentz violation based on a modified Lagrangian. We derive the covector
parametrization of the skewon optic tensor and discuss its $U(1)$-gauge
symmetry. We obtain several compact expressions for the contribution of the
principle and skewon optic tensor to the dispersion relation. As an application
of the technique proposed here, we consider the case of a generic skewon tensor
contributed to a simple metric-type principle part. Our main result: Every
solution of the skewon modified Minkowski dispersion relation is necessary
spacelike or null. It provides an extreme violation of the Lorentz symmetry.
The case of the antisymmetric skewon is studied in detail and some new special
cases (electric, magnetic, and degenerate) are discovered. In the case of a
skewon represented by a symmetric matrix, we observe a parametric gap that has
some similarity to the Higgs model. We worked out a set of specific examples
that justify the generic properties of the skewon models and demonstrate the
different types of the Lorentz violation phenomena.
