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">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 J.C. Kluyver in 1906, and also referred to as Rayleigh's theorem) for the probability $p_n$ to return to the unit disc after $n$ steps,
$$p_n=\int_0^1 P_n(r)\,dr=\frac{1}{n+1},$$
see example 2.3 in <A HREF="https://arxiv.org/abs/1508.04729">arXiv:1508.04729</A>. A purely geometric derivation is given in <A HREF="https://arxiv.org/abs/1007.4870">(Very) short proof of Rayleigh's Theorem (and extensions)</A>.