We introduce and study an alternative form of the chaotic expansion for Poisson functionals, using a very specific (non-equivalent) Girsanov transformation; we name this alternative form pseudo-chaotic expansion. This pseudo-chaotic expansion is derived on a general phase space and we analyze the link between this expansion and the classical chaotic decomposition available for random Poisson measures. When combined with the Poisson imbedding representation for point processes this expansion provides a new decomposition for this process and in the specific case of a linear Hawkes process, the coefficients of the pseudo-chaotic expansion are derived in closed form whereas those of the classical chaotic decomposition are defined through the distribution function of the marginal laws of the Hawkes process or of its intensity that cannot be obtained explicitly. Finally, this expansion enables us to study further the structure of linear Hawkes processes by constructing an example of a process in a pseudo-chaotic form that satisfies the stochastic self-exciting intensity equation which determines a Hawkes process (in particular its expectation equals the one of a Hawkes process) but which fails to be a counting process.