Covariant Lyapunov vectors characterize the directions along which perturbations in dynamical systems grow. They have also been studied as predictors of critical transitions and extreme events. For many applications, it is necessary to estimate these vectors from data since model equations are unknown for many interesting phenomena. We propose an approach for estimating covariant Lyapunov vectors based on data records without knowing the underlying equations of the system. In contrast to previous approaches, our approach can be applied to high-dimensional datasets. We demonstrate that this purely data-driven approach can accurately estimate covariant Lyapunov vectors from data records generated by several low- and high-dimensional dynamical systems. The highest dimension of a time series from which covariant Lyapunov vectors are estimated in this contribution is 128.