In the present paper, we provide cohomology groups for matroids, as a categorification of the characteristic polynomial of matroids. The construction depends on the ``quasi-representation'' of a matroid. For a certain choice of the quasi-representation, we show that our cohomology theory gives a generalization of the chromatic cohomology introduced by L. Helme--Guizon and Y. Rong, and also the characteristic cohomology introduced by Z. Dancso and A. Licata.