  It is fairly well known that rotation in three dimensions can be expressed as
a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A
generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric
generating matrix is derived for the rotation matrix of tensors of order $n$.
The Euler-Rodrigues formula for rigid body rotation is recovered by n=1. A
Cayley form of the n-th order rotation tensor is also derived. The
representations simplify if there exists some underlying symmetry, as is the
case for elasticity tensors such as strain and the fourth order tensor of
elastic moduli. A new formula is presented for the transformation of elastic
moduli under rotation: as a 21-vector with a rotation matrix given by a
polynomial of degree 8. Explicit spectral representations are constructed from
three vectors: the axis of rotation and two orthogonal bivectors. The tensor
rotation formulae are related to Cartan decomposition of elastic moduli and
projection onto hexagonal symmetry.
