This work analyses a class of extensions of the Standard Model through the lens of a novel exact parameterisation. This parameterisation is specially suitable to the analysis of extensions of the Standard Model with non-unitary mixing matrices, like models with vector-like fermions or right-handed neutrinos. The usefulness of this parameterisation is motivated with two example models: A) Standard Model with the addition of $n_u$ up and $n_d$ down singlet vector-like quarks and B) Standard Model with the addition of $n_R$ right-handed neutrinos.