Differential forms as tensors We can define differential k-forms at point $p$ of manifold $M$ to be elements of $\Lambda^k(T^*_pM)$. However, as $\Lambda^k(T^*_pM)\subset\Lambda(T^*_pM)=\frac{T(T^*_pM)}{I}$ elements of k-th exterior power are equivalence classes.
Question: How do we make tensors our of this equivalence classes?  
I know there is a map: $i:\Lambda^k(V)\to T^k(V)$
$[v_1] \wedge\dots\wedge [v_k]\mapsto\sum_\sigma(sgn(\sigma))v_{\sigma(1)}\otimes\dots \otimes v_{\sigma(k)}$
But is this map chooses representative from each class uniquely? I mean you can always pick from the class arbitrary representative, how do you know which one should you pick for this map to well-defined.
For example: we can look at particular element of $\Lambda^2(V)$, say this one: $[a\otimes b+c\otimes d+e\otimes f]$. How do I extract exact tensor from this class?
 A: For any vector space $V$, there is a canonical map $$\textrm{Alt} : T^k V^* \to T^k V^*$$ defined by 
$$(\textrm{Alt}\, \phi)(X_1, \ldots, X_k) := \frac{1}{k!} \sum_{\sigma \in S_k} (\textrm{sgn}\, \sigma) \phi(X_{\sigma(1)}, \ldots X_{\sigma(k)}) .$$ For any $\phi$, $\textrm{Alt}\,\phi$ is alternating; a $k$-form form $\psi$ is alternating iff for all $\sigma \in S_k$ we have
$$\psi(X_{\sigma(1)}, \ldots, X_{\sigma(k)}) = (\textrm{sgn}\,\sigma) \psi(X_1, \ldots, X_k) .$$
With this definition, we can give the characterization you ask about: Under your identification of $\Lambda^k V^*$ as a quotient of $T^k V^*$, checking the definition shows that $i(\omega)$ is alternating, so $i$ maps an element of $\Lambda^k V^*$ to the unique representative $\phi$ of $\Lambda^k V^*$ in $T^k V^*$ such that $\phi$ is alternating, that is, such that $\textrm{Alt}\,\phi = \phi$. Since $i$ is injective, we may identify $\Lambda^k V^*$ with $\textrm{im}\, i$, and hence as a vector subspace of $T^k V^*$, which is how we usually regard it.
One can check that $\textrm{Alt}$ is idempotent, that is, that $\textrm{Alt} \circ \textrm{Alt} = \textrm{Alt}$ (it is, in fact, a vector space projection in $T^k V^*$), so to find the representative $i([\omega])$ of a class $[\omega]$, we can simply compute $\textrm{Alt} \, \omega$ for any representative $\omega \in [\omega]$. Applying the definition for $k = 2$ gives
$$(\textrm{Alt}\, \omega)(X_1, X_2) = \frac{1}{2}(\omega(X_1, X_2) - \omega(X_2, X_1)),$$ or, for $\alpha, \beta \in V^*$,
$$\textrm{Alt}(\alpha \otimes \beta) = \frac{1}{2}(\alpha \otimes \beta - \beta \otimes \alpha) .$$
(I should add that the above formulation must be modified if the field underlying $V$ has positive characteristic $\leq k$. In that case, $k! = 0$, so our formula for $\textrm{Alt}$ and the rest of our treatment must be modified.)
