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In writing down 4D relativistic field theories, we need to choose fields that have enough degrees of freedom to accommodate the degrees of freedom of the particles we want in our theory. So, if we want to describe the physics of a massive spin-1 particle, we want to use a 4-vector field $A_\mu$, since this is the 'smallest' field that can accommodate the three degrees of freedom of our particle.

In describing the physics of a spin-2 particle (which in general has five degrees of freedom), the starting point seems to be a symmetric tensor field $h_{\mu \nu}$. This has ten degrees of freedom. My question is:

Why not embed a spin-2 particle in an antisymmetric tensor field? This has only six degrees of freedom. Removing one degree of freedom seems like it would be easier than removing the five from a symmetric tensor, necessary to reach the five degrees of freedom of a spin-2 particle.

More generally, I've heard that it's often possible to formulate a theory with given particle content in multiple ways, as far as field content is concerned. That is, one can choose to embed a spin-1 particle in a 4-vector field $A_\mu$, but also in a 3-form field $B_{\mu \nu \rho}$ if one desires (that example is illustrative and may not be correct in actuality). My second question is:

Does anybody have a reference which discusses this possibility? For instance, does anybody have some concrete examples of two or more field theories involving different types of field, but which nevertheless describe the same particles interacting in the same way?

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An antisymmetric tensor field $B_{\mu\nu}=-B_{\nu\mu}$ in 3+1D corresponds to the 6-dimensional reducible Lorentz representation $(1,0)\oplus (0,1)$, which has no spin-2 components. See also this related Phys.SE post.

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