isn't that a matter of terminology? i.e. exterior forms are often referred to as "alternating tensors".
i.e. for any module or vector space, its exterior algebra is a quotient of the tensor algebra by the homogeneous ideal generated by all squares of elements of degree one. no?
so in this sense an exterior form is an equivalence class of tensors.
there are skew - symmetric and well as symmetric tensors.
indeed if we think dually of a tensor as a field of multilinear functionals on the tangent spaces, as is usually done in differential geometry, then adding a condition like skew symmetry, merely restricts the class of tensors to a subclass.
in this sense antisymmetric tensors are actually a subfamily of the general tensors, hence they are actual tensors, and not just equivalence classes of them.