Let $(\theta,A\theta)=\theta_i A_{ij}\theta_j$ where $A$ is some $(2\times2)$ antisymmetric matrix.
I want to generalize the following
$$I(A) =\int d\theta_1d\theta_2~ \exp\Bigg[\frac{1}{2}(\theta,A\theta)\Bigg]=\int d\theta_1d\theta_2~ (1+\theta_1\theta_2A_{12}) = A_{12}=\sqrt{\det A}$$
to the $n$-tuple case.
Let now $$A:=\begin{bmatrix} 0 & 1 & \; & \; \\ \;-1 & 0 & & \; \\ \; & \; & 0 & 1 \\ \; & \; & -1 & 0 \\ \; & \; & \; & \, &\ddots \\ \end{bmatrix}.$$
I evaluate, I get the following $$I(A) = \int d\theta_n\dots d\theta_1\,\exp\Bigg[\frac{1}{2}(\theta,A\theta)\Bigg]\\ = \int d\theta_n\dots d\theta_1\, (\theta_1\theta_2+\theta_3\theta_4+\cdots)\\=0$$
The answer should be $$I(A) = 1.$$
In the above I use (perhaps incorrectly?)
$$\int d\theta_n\dots d\theta_1\, \theta_n\dots \theta_1\, = 1 $$ and $$\int d\theta_n\dots d\theta_1= 0. $$
Where do I err?
EDIT: I think I know how to fix this: it is the last term in the expansion of the exponential the contributes. All other terms give zero (just like the one above). I will add the solution later.