**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 $J_0$,
$$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$
$$F_n(r)\equiv\int_0^r P_n(r')\,dr'=\int_0^\infty [J_0(k/r)]^nJ_1(k )\,dk,\;\;0<r<n,$$
see theorem 2.1 in <A HREF="https://arxiv.org/abs/1508.04729">arXiv:1508.04729</A> for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in <A HREF="https://scholarship.claremont.edu/jhm/vol6/iss1/7/">A Short Walk can be Beautiful</A>.]

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 <A HREF="https://www.dwc.knaw.nl/DL/publications/PU00013859.pdf">J.C. Kluyver in 1905</A>) for the probability $p_n$ to return to the unit disc after $n$ steps,
$$p_n=F_n(1)=\frac{1}{n+1},\;\;n>1.$$
A purely geometric derivation is given in <A HREF="https://arxiv.org/abs/1007.4870">arXiv:1508.04729</A>.

For $n\gg 1$ we may approximate $J_0(k/r)^n\approx\exp\left(-\frac{nk^2}{4r^2}\right)$, see page 345 of Kluyvers paper, to obtain
$$F_n(r)\approx\int_0^\infty \exp\left(-\frac{nk^2}{4r^2}\right)J_1(k)\,dk=1-e^{-r^2/n},$$
which gives the expected Maxwell distribution for $P_n(r)=dF_n(r)/dr$.