Gaussian integral with respect to Grassmann variables

Let $A$ be an antisymmetric matrix of even dimension $n$ and $theta$ be a column vector consisting of $n$ Grassmann variables $theta_i$. Then the solution of the integral
$$int dtheta_1dots dtheta_n exp(-frac{1}{2}theta^T Atheta)tag{1}$$
is given by ${rm Pf} (A)$.
My question is: What is the solution of the integral if it is not assumed that $A$ is antisymmetric.

The origin of my question is a derivation in "Geometry, Topology and Physics" by Nakahara. Nakahara says the solution of
$$prod_{k=1}^Nint dtheta_k^*dtheta_k ,e^{-theta^dagger B_N theta}tag{2}$$
where
$$theta=begin{pmatrix}theta_1vdotstheta_Nend{pmatrix},,theta^dagger=begin{pmatrix}theta_1^*,dots,theta_N^*end{pmatrix},,B_N=begin{pmatrix} 1 & 0 & dots & 0 & -y y & 1 & 0 &dots 0 & 0 & y & 1 & dots 0& vdots & & ddots & ddots & vdots &0&dots&y&1 end{pmatrix}tag{3}$$
is given by $det B_N$. However $B_N$ is not antisymmetric. How does Nakahara solve this integral?