Compression optical coherence elastography forms images based on the mechanical properties of tissue by mapping the local strain in response to a compressive load. Strain is described by a second-order tensor, comprising six independent components. The majority of compression elastography methods, however, measure and form their analyses from a single axial component, which relies on the assumption that the stress in tissue is uniform and uniaxial. However, in general, tissues are complex and heterogeneous, and rarely comply with this assumption. This can lead to inaccuracies and misinterpretation of image contrast in compression optical coherence elastography. Here, we image the full strain tensor and demonstrate its utility in unambiguously characterizing deformation in structured phantoms and ex vivo tissues. We derive additional parameters from the strain tensor, and map the local compressibility, anisotropy of deformation, and total equivalent strain. Such analysis is enabled by an efficient non-iterative approach to measuring the three-dimensional displacement field via a closed-form solution to a collection of the amplitude of complex correlation coefficients across multiple digitally shifted images. Strain tensor imaging is likely to lead to more accurate estimation of tissue mechanical properties, improving the utility of compression optical coherence elastography in clinical and biological applications.