It's not difficult to see that the graviton in $D$ spacetime dimensions has $(D-3)D/2$ polarizations. In $D=4$ there are two $\epsilon^{\pm}_{\mu\nu}$. What I find curious is that in $D=4$ I can actually pick $\epsilon^{\pm}_{\mu\nu}=\epsilon^{\pm}_{\mu}\epsilon^{\pm}_{\mu}$$\epsilon^{\pm}_{\mu\nu}=\epsilon^{\pm}_{\mu}\epsilon^{\pm}_{\nu}$ where $\epsilon^{\pm}_{\mu}$ are the two polarizations (of definite elicity $\pm1$) for a massless spin-1 particle like the photon. In higher dimension this doesn't seem possible since the photon has $D-2$ polarizations so that the number $(D-2)(D-1)/2$ of $\epsilon^{\lambda}_{\mu}\epsilon^{\lambda^\prime}_{\mu}$ pairs doesn't match the number $(D-3)D/2$ of graviton polarization. Well, unless somehow I consider only a smaller subset of them, say adding a constraint or removing one of them $$(D-2)(D-1)/2-1=(D-3)D/2$$ as in $D=4$ where $\epsilon^{+}_{\mu}\epsilon^{-}_{\mu}$ is discarded having zero elicity.
Is there an analogous constructions for $\epsilon^{\lambda}_{\mu\nu}$ in terms of the spin-1 polarizations $\epsilon^\sigma_\mu$ in higher dimensions? I suspect that something similar may happen only if I involve polarizations of higher elicity as well.
