For the record, to answer the question in the OP: The desired PDF $P_n(r)$ of the distance $r$ from the origin for arbitrary $n$ 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  <A HREF="https://arxiv.org/abs/1508.04729">Densities of short uniform random walks in higher dimensions</A> for a derivation (and the generalization to $d$ dimensions). 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 $1/(n+1)$ for the probability to return to the unit disc after $n$ steps [Kluyver's theorem, see example 2.3].