[This answer is still a work in progress. I'll get back to it later]. You can calculate an expression for this, for any n. Let your volume be V.
By scaling, the volume of the set {|A|≤K} is VKn2. Now let M be a matrix whose entries are independent normal random variables with mean 0 variance 1. From the density function of the normal distribution, this gives P(|M|≤K)~(2π)-n2/2VKn2 in the limit of small K.
I'll now calculate this expression in an alternative way. Use the M=QR decomposition, where Q is orthogonal and R is upper triangular, with diagonal elements λn, λn-1,…λ1. This can be done in such a way that λk2 has the χ2k-distribution (a quick google search gives this but there's probably better references). The upper triangular parts of R have the standard normal density. I have to think how we can calculate the |R|. I was originally thinking that this is the max eigenvalue, but it's not quite that simple.