Spectroscopic ellipsometry measurements obtained on three differently oriented surfaces of single crystalline 3,4,9,10-perylene tetracarboxylic dianhydride (alpha-PTCDA) are combined and analyzed according to the symmetry of the monoclinic crystal. In this scheme, the exact ellipsometric equations are solved numerically within a 4 x 4 transfer matrix algorithm. From the analysis of the data the three principal components of the complex dielectric tensor and their orientation with respect to the crystal are determined as function of energy. In this work, we report accurate values of the real dielectric tensor which are expected to be useful for optical applications of PTCDA in the transparency range of the material.