We study structure and direct decay of the Giant Monopole Resonance (GMR) at the RPA level using the Time-Dependent Energy Density Functional method in the linear response regime in a few doubly-magic nuclei. A proper treatment of the continuum, through the use of large coordinate space, allows for a separation between the nucleus and its emitted nucleons. The microscopic structure of the GMR is investigated with the decomposition of the strength function into individual single-particles quantum numbers. A similar microscopic decomposition of the spectra of emitted nucleons by direct decay of the GMR is performed. In this harmonic picture of giant resonance, shifting every contribution by the initial single-particle energy allows to reconstruct the GMR strength function. The RPA residual interaction couples bound 1-particle 1-hole states to unbound ones, allowing for the total decay of the GMR. In this article, we then intend to get an understanding of the direct decay mechanism from coherent one-particle-one-hole superpositions, while neglecting more complex configurations. Time-dependent beyond mean-field approaches should be use, in the future, to extend this method.