As everyone above has pointed out, the expected value is $0$.

I expect that the original poster might have wanted to know about how big the determinant is. A good way to approach this is to compute $\sqrt{E((\det A)^2)}$, so there will be no cancellation.

Now, $(\det A)^2$ is the sum over all pairs $v$ and $w$ of permutations in $S_n$ of 
$$(-1)^{\ell(v) + \ell(w)} (1/2)^{2n-\# \{ i : v(i) = w(i) \}}$$

Group together pairs $(v,w)$ according to $u := w^{-1} v$. We want to compute
$$(n!) \sum_{u \in S_n} (-1)^{\ell(u)} (1/2)^{2n-\# (\mbox{Fixed points of }i)}$$

This is $(n!)^2/2^{2n}$ times the coefficient of $x^n$ in 
$$e^{2x+x^2/2+x^3/3 + x^4/4 + \cdots} = e^x/(1-x).$$

So $\sqrt{E((\det A)^2)}$ is
$$\sqrt{(n!)^2/2^{2n} \left( 1+1/2+1/6+1/24+\cdots + 1/n! \right)} \approx n! \sqrt{e}/ 2^n$$