  In this paper we will develop a family of non-conforming "Crouzeix-Raviart"
type finite elements in three dimensions. They consist of local polynomials of
maximal degree $p\in\mathbb{N}$ on simplicial finite element meshes while
certain jump conditions are imposed across adjacent simplices. We will prove
optimal a priori estimates for these finite elements.
  The characterization of this space via jump conditions is implicit and the
derivation of a local basis requires some deeper theoretical tools from
orthogonal polynomials on triangles and their representation. We will derive
these tools for this purpose. These results allow us to give explicit
representations of the local basis functions. Finally we will analyze the
linear independence of these sets of functions and discuss the question whether
they span the whole non-conforming space.
