It is said (e.g. Lovelock and Rund) that p-vectors are equivalent to skew-symmetric tensors. However, a skew-symmetric form, such as $A_{ijk}$, is a specific form on $\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}^n$, while an exterior algebra $$\bigwedge\nolimits^3\mathbb{R}^n$$ only corresponds to a specific form only after a functional a chosen on it. How do you reconcile the two?
In particular, how do you introduce an inner product on skew-symmetric tensors that corresponds to the usual definition on exterior algebras (determinant of the matrix of pairwise inner products of the elements in the p-blade).