We present a full-range Fourier-domain optical coherence tomography (OCT) system that is capable of acquiring two-dimensional images of living tissue in a single shot. By using line illumination of the sample in combination with a two-dimensional imaging spectrometer, 1040 depth scans are performed simultaneously on a sub-millisecond timescale. Furthermore, we demonstrate an easy and flexible real-time single-shot technique for full-range (complex-conjugate cancelled) OCT imaging that is compatible with both two-dimensional as well as ultrahigh-resolution OCT. By implementing a dispersion imbalance between reference and sample arms of the interferometer, we eliminate the complex-conjugate signal through numerical dispersion compensation, effectively increasing the useful depth range by a factor of two. The system allows us to record 6.7 x 3.2 mm images at 5 microm depth resolution in 0.2 ms. Data postprocessing requires only 4 s. We demonstrate the capability of our system by imaging the anterior chamber of a mouse eye in vitro, as well as human skin in vivo.