What is "two-vector"

A two-vector is a tensor of type (2,0) and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars).

The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then

$$\mathbf{f} = f^{\alpha \beta} \, \vec e_\alpha \otimes \vec e_\beta$$
where the f are the components of the two-vector. Notice that both indices of the components are contravariant. This is always the case for two-vectors, by definition.

An example of a two-vector is the inverse g of the metric tensor.

The components of a two-vector may be represented in a matrix-like array. However, a two-vector, as a tensor, should not be confused with a matrix, since a matrix is a linear function

M : V → V
which maps vectors to vectors, whereas a two-vector is a linear functional

f : Ṽ → V
which maps one-forms to vectors. In this sense, a matrix, considered as a tensor, is a mixed tensor of type (1,1) even though of the same rank as a two-vector.