In this paper we study the local description of spaces of forms on transitive
Lie algebroids. We use this local description to introduce global structures
like metrics, ∗-Hodge operation and integration along the algebraic part
of the transitive Lie algebroid (its kernel). We construct a \v{C}ech-de Rham
bicomplex with a Leray-Serre spectral sequence. We apply the general theory to
Atiyah Lie algebroids and to derivations on a vector bundle