Let $\mathbf{T}$ be (for example) a rank-2 antisymmetric covariant tensor, with components $T_{ij}$. In the language of differential forms, we can represent $\mathbf{T}$ as $$ \mathbf{T}=\sum_{i,j}\frac{1}{2}\,T_{ij}\,\mathrm{d}x^i\wedge\mathrm{d}x^j, $$ and similarly for higher ranks.
Is there an equivalent way to express a symmetric tensor? What sort objects would it be built out of?
In particular, we know that any tensor can be written as the sum of its symmetric and antisymmetric parts. If we drop the requirement that $\mathbf{T}$ be antisymmetric, then using the above notation we'd have something like $$ \mathbf{T}=\sum_{i,j}\frac{1}{2}\,T_{[ij]}\,\mathrm{d}x^i\wedge\mathrm{d}x^j + \sum_{i,j}\frac{1}{2}\,T_{(ij)}\,\left[\,\text{something other than}\; \mathrm{d}x^i\wedge\mathrm{d}x^j\,\right]. $$ What sort of object is that "something other than $\mathrm{d}x^i\wedge\mathrm{d}x^j$"? What space do the $\mathrm{d}x^i$s and not-$\mathrm{d}x^i$s collectively live in?
For background, I came to this question in the context of electromagnetism. The equations governing electromagnetism can be expressed extremely concisely and elegantly in the language of differential forms, but important tensors (namely the stress-energy tensor) are symmetric, and therefore seemingly can't play nice with that formulation. What's missing to resolve this (apparent) conflict?