We use the T-matrix approach and the analytical orientation-averaging technique to formulate the problem of light scattering by an ensemble of rotationally symmetric particles in arbitrary orientation. The mathematical formulation yields analytical expressions for the elements of the ensemble-averaged scattering matrix that involve no more than four nested summations. An expansion into generalized spherical functions is used in the particular case where the scatterers are partially aligned along the direction of incidence. A computer code that implements the analytical expressions derived is publicly available on the World Wide Web at http://irctr.et.tudelft.nl/ -Skaropoulos/T-matrix.htm.