  We construct via generators and relations a generalized Weil representation
for the split orthogonal group $\ot_q(2n,2n)$ over a finite field of $q$
elements. Besides, we give an initial decomposition of the representation
found. We also show that the constructed representation is equal to the
restriction of the Weil representation to $\ot_q(2n,2n)$ for the reductive dual
pair $(\Sp_2(\mathbb{F}_q),\ot_q(2n,2n))$ and that the initial decomposition is
the same as the decomposition with respect to the action of
$\Sp_2(\mathbb{F}_q)$.
