Dirac-Kähler approach connected to quantum mechanics in Grassmann space

We compare the way one of us obtained spinors from fields, which are a priori antisymmetric tensor fields, to the Dirac-K\"ahler rewriting. Since using our Grassmann formulation is simple it may be useful in describing the Dirac-K\"ahler formulation of spinors and in generalizing it to vector internal degrees of freedom and to charges. The ``cheat'' concerning the Lorentz transformations for spinors is the same in both cases and is put forward in the Grassmann formulation. Also the generalizations are clearly pointed out. The discrete symmetries are discussed, in particular the appearance of two kinds of time-reversal operators as well as the unavoidability of four families and the evenness and oddness of the Dirac matrices.