Holding promising applications in power electronics, the wide band gap material gallium oxide has emerged as a vital alternative to materials like GaN and SiC. The detailed study of phonon modes in $beta$-Ga$_{2}$O$_{3}$ provides insights into fundamental material properties such as crystal structure and orientation and can contribute to the identification of dopants and point defects. We investigate the Raman active phonon modes of $beta$-Ga$_{2}$O$_{3}$ in two different oxygen isotope compositions ($^{16}$O,$^{18}$O) by experiment and theory: By carrying out polarized micro-Raman spectroscopy measurements on the (010) and ($bar{2}$01) planes, we determine the frequencies of all 15 Raman active phonons for both isotopologues. The measured frequencies are compared with the results of density functional perturbation theory (DFPT) calculations. In both cases, we observe a shift of Raman frequencies towards lower energies upon substitution of $^{16}$O with $^{18}$O. By quantifying the relative frequency shifts of the individual Raman modes, we identify the atomistic origin of all modes (Ga-Ga, Ga-O or O-O) and present the first experimental confirmation of the theoretically calculated energy contributions of O lattice sites to Raman modes. We find that oxygen substitution on the O$_{mathrm{II}}$ site leads to an elevated relative frequency shift compared to O$_{mathrm{I}}$ and O$_{mathrm{III}}$ sites. This study presents a blueprint for the future identification of different point defects in Ga$_{2}$O$_{3}$ by Raman spectroscopy.
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