In this paper, Conjugated Unscented Transformation (CUT) based approach is presented to compute higher order state transition matrices in a derivative free manner and a computationally attractive manner. The proposed approach is non-intrusive in nature and is similar to stochastic collocation methods. The connection between stochastic collocation methods, geometric series methods and the conventional higher order state transition matrix approach are discussed. The computed state transition matrices are valid over the desired domain represented by a probability density function rather than valid along a single trajectory of a dynamical system. Benchmark problems corresponding to uncertainty propagation are considered to demonstrate the numerical efficiency and accuracy of the proposed ideas.