Let us consider a linear stochastic oscillator system: $$ d \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x(t) \\y(t)\end{bmatrix}dt + \alpha \begin{bmatrix} 0 \\ 1 \end{bmatrix} dW_t$$
we know that the phase flow preserves symplectic structure.
So, given an initial conditions and a numerical method of the form $$ \begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21}& a_{22} \end{bmatrix} \begin{bmatrix} x_n\\ y_n \end{bmatrix}+ \alpha \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \Delta W_n$$
we have that the method preserves the symplectic structure iff $\det(A)=1$, where $A$ is the matrix that appears in the numerical scheme with entries $a_{ij}$
I need to impose $d x(t) \wedge d y(t) = d x_0 \wedge dy_0$, but I really don't know how to do
The problem with the notation is that $d$ in the first version describes an (infinitesimal) increment relative to time, while in the last equation $d$ is the (also infinitesimal) displacement in space. Use subscripts to make a distinction, $d_t$ and $d_s$.
Then for the exact equation you get $$ d_t(d_sx)=d_s(d_tx)=d_s(ydt)=d_sy\,dt\\ d_t(d_sy)=d_s(d_ty)=d_s(-xdt+αd_tW)=-d_sx\,dt\\ d_t(d_sx\wedge d_sy)=d_t(d_sx)\wedge d_sy+d_sx\wedge d_t(d_sy)=0 $$ You do not get an extra Ito term because 1) $dW$ is only present in one factor and 2) its contribution is constant under space displacement. Thus $d_sx\wedge d_sy$ is constant.
The same essentially applies if you compute the evolution of $d_sx_n\wedge d_sy_n$.