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Skyrmions are small topologically protected magnetic structures that hold promise for applications from data storage to neuromorphic computing and they have been shown to possess internal microwave frequency excitations. Skyrmions in a synthetic antiferromagnet have been predicted to be smaller and faster than their ferromagnetic equivalents and also shown to possess more internal modes. In this work, we consider the breathing modes of skyrmions in a four repetition synthetic antiferromagnetic multilayer by means of micromagnetic simulations and examine the further splitting of the modes into different arrangements of out-of-phase, in-phase, and modes with more complex phase relationships. This results in a lowering of frequencies, which is promising for skyrmion sensing applications in a synthetic antiferromagnet. [ABSTRACT FROM AUTHOR]