The measurement of a quantum state poses a unique challenge for experimentalists. Recently, the technique of 'direct measurement' was proposed for characterizing a quantum state in situ through sequential weak and strong measurements. While this method has been used for measuring polarization states, its real potential lies in the measurement of states with a large dimensionality. Here we show the practical direct measurement of a high-dimensional state vector in the discrete basis of orbital angular momentum. Through weak measurements of orbital angular momentum and strong measurements of angular position, we measure the complex probability amplitudes of a pure state with a dimensionality, d = 27. Further, we use our method to directly observe the relationship between rotations of a state vector and the relative phase between its orbital-angular-momentum components. Our technique has important applications in high-dimensional classical and quantum information systems and can be extended to characterize other types of large quantum states.