  In this paper we show that every rational cohomology class of type $(p,p)$ on
a compact K\"ahler manifold can be representated as a differential $(p,p)$-form
given by an explicit formula involving a \v{C}ech cocycle. First we represent
Chern characters of smooth vector bundles by \v{C}ech cocycles with values in
the sheaf of differential forms. We then consider the behavior of these
cocycles with respect to the Hodge structure on cohomology when the base
manifold is compact K\"ahler.
