In 4D spacetime, spin charactarizes the transformation properties of fields under the transformation of the little group, which is $SO(3)$ for massive particles and $SO(2)$ for massless ones (if you are not familiar with the little groups, I recommend you to check Ch.2 of Weinberg's course). Representations of the former one are characterized by the value of the Casimir operator of $SO(3)$, namely $l_x^2+l_y^2+l_z^2=l(l+1)$, here $l$ is called spin. For the case of massless particles, the situation is even simpler -- transformation properties are governed by the single number $\mathrm{e}^{\mathrm{i} m \phi}$, which multiplies the wave-function when one rotates the coordinate system about the direction in which particle travels (here $m$ is what is called spin).

In higher dimensions the situation is similar, but now one has to consider representations of $SO(D-1)$ and $SO(D-2)$ groups for massive and massless particles respectively. In general, these groups have more than one Casimir (for example, $\mathfrak{so}(4)=\mathfrak{so}(3) \times \mathfrak{so}(3)$, so it has two Casimirs), and the representation is characterized by the values of all these Casimirs (there is a corresponding "spin" of each of them). Usually, spin for these cases is extracted from the value of the largest of them. Being defined this way, it has zero value for scalars, 1 -- for vectors, 2 -- for gravitons, and 1 -- for all the higher rank antisymmetric fields.

See the discussion on construction of general representations in terms of spinors of $SU(2)$ in the Appendix to the second volume of Polchinski's "String Theory".

This post imported from StackExchange Physics at 2017-01-30 14:42 (UTC), posted by SE-user Andrew Feldman