The Bell basis is a distinctive set of maximally entangled two-particle quantum states that forms the foundation for many quantum protocols such as teleportation, dense coding, and entanglement swapping. While the generation, manipulation, and measurement of two-level quantum states are well understood, the same is not true in higher dimensions. Here we present the experimental generation of a complete set of Bell states in a four-dimensional Hilbert space, comprising 16 orthogonal entangled Bell-like states encoded in the orbital angular momentum of photons. The states are created by the application of generalized high-dimensional Pauli gates on an initial entangled state. Our results pave the way for the application of high-dimensional quantum states in complex quantum protocols such as quantum dense coding.