Let $(x^1, \ldots, x^n)$ be local coordinates on a manifold $M$. Since we are working locally, we may assume that these are global coordinates. Let us write $\partial_1, \ldots, \partial_n$ for the vector fields corresponding to these coordinates; this is a global frame (i.e. a trivialisation) of the tangent bundle $T M$. The differential 1-forms $d x^1, \ldots, d x^n$ are defined at first to be the duals of $\partial_1, \ldots, \partial_n$, so that $d x^i (\partial_j) = \delta^i_j = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \ne j \end{cases}$ and this holds pointwise.
Now, we define what $d x^{i_1} \wedge \cdots \wedge d x^{i_p}$ does. It is completely determined by the following: $(d x^{i_1} \wedge \cdots \wedge d x^{i_p}) ( \partial_{j_1}, \ldots, \partial_{j_p} ) = \begin{cases} +1 & \text{if } (i_1, \ldots, i_p) \text{ is an even permutation of } (j_1, \ldots, j_p) \\ -1 & \text{if } (i_1, \ldots, i_p) \text{ is an odd permutation of } (j_1, \ldots, j_p) \\ 0 & \text{otherwise} \end{cases}$ This can be extended by linearity to all differential $p$-forms. Note that this gives an embedding of $\Lambda^p T^* M$ into $(T^* M)^{\otimes p}$ by $d x^{i_1} \wedge \cdots \wedge d x^{i_p} \mapsto \sum_{\tau \in S_p} \textrm{sgn}(\sigma) \, d x^{i_{\tau (1)}} \otimes \cdots \otimes d x^{i_{\tau (p)}}$ and in the case $p = 2$ this amounts to $dx^{i_1} \wedge dx^{i_2} \mapsto dx^{i_1} \otimes dx^{i_2} - dx^{i_2} \otimes dx^{i_1}$ as others have said.
If $\sigma$ is a 2-form, then $\sigma = \sum \sigma_{i j} \, dx^i \wedge dx^j$ for some smooth functions $\sigma_{i j}$... except what they are depends on the convention you use. Physicists typically use the convention where $\sigma_{i j}$ is defined as $-\sigma_{j,i}$ for $i \ge j$, so that $\sigma = \sum_{i < j} \sigma_{i,j} \, dx^i \wedge dx^j = \frac{1}{2} \sum_{i, j} \sigma_{i j} \, dx^i \wedge dx^j$ Note that this is compatible with the identification of $dx^i \wedge dx^j$ with $dx^i \otimes dx^j - dx^j \otimes dx^i$. I'll use this convention here. Let $X = \sum X^i \partial_i$ and $Y = \sum Y^j \partial_j$. Then, $\sigma (X, Y) = \frac{1}{2} \sum_{i, j, k, \ell} \sigma_{i j} X^k Y^\ell \, (dx^i \wedge dx^j) (\partial_k, \partial_\ell) = \frac{1}{2} \sum_{i, j} \sigma_{i j} (X^i Y^j - X^j Y^i)$ but since $\sigma_{i j} = - \sigma_{j i}$, we can simplify the RHS to $\sigma (X, Y) = \sum_{i, j} \sigma_{i j} X^i Y^j$
More generally, if $\sigma$ is a $p$-form with $\sigma = \frac{1}{p !} \sum_{i_1, \ldots, i_p} \sigma_{i_1 \ldots i_p} \, dx^{i_1} \wedge \cdots \wedge dx^{i_p}$ and $X_1, \ldots, X_p$ are vector fields with $X_r = \sum_j X_r^j \, \partial_j$, we have $\sigma (X_1, \ldots, X_p) = \sum_{i_1, \ldots, i_p} \sigma_{i_1 \ldots i_p} X^{i_1} \cdots X^{i_p}$ where we have assumed that $\sigma_{i_1 \ldots i_p}$ is a totally antisymmetric in the sense that $\sigma_{i_{\tau (1)} \ldots u_{\tau (p)}} = \textrm{sgn}(\tau) \, \sigma_{i_1 \ldots i_p}$