The paper discusses a new idea of the multi-dimensional characteristic ellipsoid (CELL) approach to monitor dynamic behavior of an interconnected power system using phasor measurement unit (PMU) measurement. The multi-dimensional minimum volume enclosing ellipsoid (MVEE) is extracted based on phasor measurements to represent the feature of dynamic behavior of power systems. The orientation matrix of MVE describes the shape and orientation of the CELL. Extreme events can be identified by tracking the volume change of the CELL with respect to time. Some feature indexes that can be used to determine the disturbance are also proposed to interpret the dynamic behavior of the power system, and to determine the decision trees for extreme event identification procedure. This method will be a useful tool for providing wide-area situational awareness for grid operators, identification of system disturbances and detection of system stresses and their locations.