Well, there is certainly not a Majorana representation, since any decomposition will have two terms which differ by a phase of -1, you can't find Majorana points. The singlet state is anti-symmetric, so there is no way as writing it in the form
$\frac{e^{i\alpha}}{\sqrt{2}} \sum_{j=0}^1 | \phi_{1\oplus j} \rangle \otimes | \phi_{0\oplus j} \rangle$ since it is always in the state $\frac{e^{i\alpha}}{\sqrt{2}} \sum_{j=1}^2 (-1)^j| \phi_{1\oplus j} \rangle \otimes | \phi_{0\oplus j} \rangle$. Here I have taken $\{\phi_{0}, \phi_{1}\}$ as a basis for the single qubit Hilbert space.

However, if you want something kinda-sorta like the Majorana representation, you can do the following. The Majorana representation is effectively treating the subsystems like bosons, and hence we are stuck working in the symmetric subspace. However, you can do the exact same thing treating the subsystems as fermions, which will the restrict you to the antisymmetric state for a Hilbert space of that dimension.

Another route would simply be to consider states which are LU equivalent to Majorana states, but I have no idea whether this is useful to you (you haven't explained exactly what you want or why you want it). If you just care about entanglement (which is a very common usage) then LU equivalence should be fine.

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