This paper is concerned with the definition of shape functions for high-order Whitney finite element spaces. The main goal in the present paper is to study higher-order Whitney complexes and to design shape functions for Whitney

$p$-elements that use only dofs associated with integration subdomains of dimension

$p$. This task is performed with the introduction of so-called small simplices, which are defined by means of a particular homothety. Degrees of Freedom are then the integrals over suitable

$p$-dimensional subdomains, linear combinations of small

$p$-simplices. It is also provided a basis for these elements on simplicial meshes and is given a geometrical localization of all degrees of freedom.