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The modeling of out-of-equilibrium many-body systems requires to go beyond the low-energy physics and local densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including the highly excited ones, contribute to the dynamics and therefore affect the system's properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical quantum particles due to the difficulty in enumerating states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the case of systems that can be mapped into free fermions. The starting point of our method is the principal component analysis of the filling matrix $F$ describing how $N$ fermions can be configured into $L$ single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of spectra given by the principal components of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions: (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.