[tex]T_x M^*[/tex] is called the cotangent space of [tex]M[/tex] at the point [tex]x[/tex]. It is the vector space of 1-forms. Higher forms are in the vector space made from tensor products of the cotangent space, so 2-forms are in [tex]T_x M^*\otimes T_x M^*[/tex].
When we write [tex]T M^*[/tex] we mean something different, but related. This is the cotangent bundle, which is the total space of the manifold [tex]M[/tex] together with the cotangent space at every point.
An electromagnetic field tensor, or Faraday tensor, F = Fuvdxvdxv is an element of the space of two forms over the field of reals, or a type [0,2] antisymmetric tensors. This is a subspace of all type [0,2] tensors, so any Faraday tensor with lower indeces is also a member of the space of type [0,2] tensors.
Sometimes the Faraday tensor is given with upper indeces. It is still antisymmetric but a member of the antisymmetric tensors over the field of reals, but with upper indeces, so is called a type [2,0] tensor.
Or it could be presented in mixed form, type [1,1]. A vector space doesn't need or involve a manifold in it's set of axioms but can, however, be identified with the tangent space of a point on a manifold, which fzero has discussed.