The series elastic actuator (SEA) has been widely used in the exoskeletons. However, its mathematical model contains both unknown parameters and states, which need to be jointly estimated online via the measurable input and output. To address this issue, an adaptive observer is developed to reconstruct both unknown model parameters and states simultaneously. In particular, a constructive approach is proposed to extract the parameter estimation error, which is used to design a new adaptive law decoupled from the observer error dynamics. A high-gain modification is also incorporated into the observer to handle the dependency on the unknown system states in the regressor. As a result, the convergence of both the observation error and the estimation error can be proved under the standard excitation condition. Moreover, since the SEA used in the exoskeletons may operate under a weak excitation due to the slow training of rehabilitation, a further tailored adaptive law with the eigenvalue decomposition method is introduced to adapt the weak excitation. Finally, the effectiveness of the proposed methods is validated via experiments on a real-world SEA test rig.