Generalization to arbitrary $n$: The desired distribution $P_n(r)$ of the distance $r$ from the origin after $n$ steps can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ see theorem 2.1 in arXiv:1508.04729 for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in A Short Walk can be Beautiful.]
For $n=2$ we recover $P_2(r)=(2/\pi)(4-r^2)^{-1/2}$. The cited paper contains closed form expressions for $n=3,4$, in terms of hypergeometric functions (eqs. 74, 79). Closed form expressions for the moments exist for any $n$.
There is also a remarkable exact result (first obtained by J.C. Kluyver in 1905) for the probability $p_n$ to return to the unit disc after $n$ steps, $$p_n=\int_0^1 P_n(r)\,dr=\frac{1}{n+1}.$$ A purely geometric derivation is given in arXiv:1508.04729.