I am currently working on some geometric aspects of higher-spin models for which there appear antisymmetric tensor coordinates

$X^{\mu\nu}=-X^{\nu\mu}$,

with $\mu,\nu=1,...,N$,

which have been introduced in some papers, such as https://arxiv.org/pdf/hep-th/0501113.pdf and https://arxiv.org/abs/1207.5683. These generalised coordinates have been employed to describe the dynamics of higher-spin fields within N-dimensional "tensorial spaces".

Basically, from my understanding, each point of a tensorial space is parametrised by an anti-symmetric $N\times N$ matrix $X$.

I would like to know if there is any rigorous mathematical way to properly define tensorial spaces and/or if there exist references in mathematical literature in which these "generalised" spaces have been already discussed in a systematic way.