The c.d.f. of a $2 \times 2$ random determinant with mutually independent normally distributed entries is derived as an infinite series. Error functions that bound the tail of this series facilitate numerical calculation. Conditions are imposed on four variable quadratic forms for this distribution to apply. A normal approximation to the distribution is suggested.
"On the Distribution of $2 \times 2$ Random Normal Determinants." Ann. Math. Statist. 29 (2) 575 - 580, June, 1958. https://doi.org/10.1214/aoms/1177706634