Elements of $\Lambda^k(V)$ I am trying to figure out how to express the elements of $\Lambda^k(V)$, but I can't really find a consistent result for this. If $\omega \in \Lambda^k(V)$, then apparently you can express it as $$\omega = \sum_{\sigma \in S(k,n-k)} \omega(e_{\sigma(1)}, \dots, e_{\sigma(k)})\varepsilon_{\sigma(1)} \wedge\dots\wedge \varepsilon_{\sigma(k)}$$ where the $e$'s are the basis for $V$ and $\varepsilon$'s are the dual base for $V$ and $\dim V=n$.
However another one I've seen is that if $\omega \in \Lambda^k(V)$, then $$\omega=\sum_{i_1<\dots<i_k} \omega_{i_1}, \dots\omega_{i_k} \varepsilon_{i_1} \wedge \dots \wedge \varepsilon_{i_k}.$$
Also alternatively we have that $\omega \in \Lambda^k(V) \subset T^k(V)$ so $\omega$ is a $k$-covariant tensor and thus $$\omega = \sum_{j_1,\dots,j_k=1}^n \omega(e_{j_1}, \dots, e_{j_k}) \varepsilon_{j_1} \otimes \dots \otimes \varepsilon_{j_k}.$$
Could I have some help on sorting out this confusion? In the last expression I see that there will be a lot of zero terms, but the two results above I don't quite get. I also don't know what this notation $\sum_{i_1<\dots<i_k}$ is supposed to mean.
 A: You mixed several notations/concepts. I'll start from the end.
First, let's recall that $\Omega^k(V)$ is the vectorial space of $k$-forms over the $n$-dimensional space $V$. As you said, it is subspace of $T^k(V^*)$, i.e. the $k$-covariant tensors, whose basis is given by the canonical dual basis $\{\varepsilon_{i_1} \otimes \ldots \otimes \varepsilon_{i_k}\}_{1 \le i_1, \ldots, i_k \le n}$, such that $\omega \in \Omega^k(V)$ can be expressed as
$$
\omega = \sum_{i_1,\ldots,i_k=1}^n a_{i_1\ldots i_k}\, \varepsilon_{i_1}\otimes\ldots\otimes\varepsilon_{i_k},
$$
where $a_{i_1\ldots i_k}$ denotes the components of $\omega$ with respect to that basis. Since $\omega$ is a linear map on the space $\Lambda^k(V) \subset T^k(V)$ of $k$-contravariant tensors, those components correspond to the image of the canonical basis of that last space, i.e. $a_{i_1\ldots i_k} = \omega(e_{i_1},\ldots,e_{i_k})$.
Next, as the exterior product is the antisymmetrized version of the tensorial product, it is alternating for the elements of the basis, so that all non-zero components must have distinct indices. Moreover, basis vectors with permuted indices are redundant du to the antisymmetry, such that one can always order them increasingly. In consequence, the basis of $\Omega^k(V)$ is given by $\{\varepsilon_{i_1} \wedge \ldots \wedge \varepsilon_{i_k}\}_{1 \le i_1<\ldots<i_k \le n}$, hence the representation
$$
\omega = \sum_{1\le i_1<\ldots<i_k \le n} \omega_{i_1\ldots i_k}\, \varepsilon_{i_1}\wedge\ldots\wedge\varepsilon_{i_k}
$$
in that basis.
Finally, the first definition you gave is an intermediate version of the two previous ones, after getting rid of the zero-components due to alternance but before reordering the basis with increasing indices by antisymmetry.
