Description
Let $N\in\mathbb{N}^{+}$ and $X_{n}\stackrel{IID}{\sim}U(0,1)$ for $n\in\{1,...,N\}$.
Given $X_{1}\leq X_{2}\leq X_{3}\leq...\leq X_{N}$, I would like to understand $f_{X_{n}}$ by writing out $f_{X_{1}}$, $f_{X_{2}}$ and $f_{X_{3}}$ (as part of choosing a proper prior for some Bayesian test).
Attempt
Given $N=2$, by letting $x_{1}=0\implies P(X_{1}\leq X_{2})=1$ such that $f_{X_{2}|X_{1}\leq X_{2}}=f_{X_{2}}=1$ and $x_{1}=1\implies P(X_{1}\leq X_{2})=0$ such that $f_{X_{2}|X_{1}\leq X_{2}}=f_{X_{2}}=0$, my educated guess would be $f_{X_{1}|X_{1}\leq X_{2}}=1-x_{1}$ and $f_{X_{2}|X_{1}\leq X_{2}}=x_{2}$ as these dependances are (probably) linear due to the underlying uniform distributions.
It seems this is confirmed by Bayes theorem as follows $$f_{X_{2}|X_{1}}=\frac{f_{X_{1}|X_{2}}f_{X_{2}}}{f_{X_{1}}}=f_{X_{1}|X_{2}}=\int^{x_{2}}_{0}dX_{1}=x_{2}$$ and $$f_{X_{1}|X_{2}}=\frac{f_{X_{2}|X_{1}}f_{X_{1}}}{f_{X_{2}}}=f_{X_{2}|X_{1}}=\int^{1}_{x_{1}}dX_{2}=1-x_{1}$$ However, for $N=3$ I am stuck as I would not know $f_{X_{1},X_{2}}$ in $$f_{X_{3}|X_{1},X_{2}}=\frac{f_{X_{1},X_{2}|X_{3}}f_{X_{3}}}{f_{X_{1},X_{2}}}$$