computation involving exterior $2$-form on $\mathbb{R}^n$

Question

Let $$\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$$be an exterior $2$-form on $\mathbb{R}^n$, and $A, B \in \mathbb{R}^n$ are vectors$$A = (1, 1, 1, \dots, 1),\text{ }B = (-1, 1, -1, \dots, (-1)^n).$$How do I compute $\theta(A, B)$?

Answer 1

We have $$\theta = \sum_{i=1}^{n-1} x_i \otimes x_{i+1} - \sum_{i=1}^{n-1} x_{i+1} \otimes x_i$$so$$\theta(A, B) = \sum_{i=1}^{n-1} A_iB_{i+1} - \sum_{i=1}^{n-1} A_{i+1}B_i = \sum_{i=1}^{n-1}(-1)^{i+1} - \sum_{i=1}^{n-1}(-1)^i = \left\{ \begin{array}{ll} 2 & \mbox{if } n \text{ even}\\ 0 & \mbox{if } n \text{ odd} \end{array} \right..$$