How to build a many-body state starting from single-particle states? Suppose that I have 3 non-degenerate single-partilce energy levels $E_1$, $E_2$, and $E_3$, each one associated to eigenstates $|\psi_1\rangle$, $|\psi_2\rangle$, and $|\psi_3\rangle$. 
How do you build the most general many-body state in the case particles are spinless fermions?
How does the answer change if, instead of fermions, particles are bosons?
 A: It is easier to do it in terms of wave functions than in the bra-ket notation. The one-particle states are $$|\psi_1\rangle\leftarrow\psi_1(x),|\psi_2\rangle\leftarrow\psi_2(x), |\psi_3\rangle\leftarrow\psi_3(x)$$. The two-particle states are antisymmetrized combinations of the pairwise products, i.e. $$|\psi_1\psi_2\rangle\leftarrow
[\psi_1(x_1)\psi_2(x_2) - \psi_1(x_2)\psi_2(x_1)]/\sqrt{2},\\|\psi_1\psi_3\rangle\leftarrow
[\psi_1(x_1)\psi_3(x_2) - \psi_1(x_2)\psi_3(x_1)]/\sqrt{2},\\|\psi_2\psi_3\rangle\leftarrow
[\psi_2(x_1)\psi_3(x_2) - \psi_2(x_2)\psi_3(x_1)]/\sqrt{2}.
$$
Finally, the three-particle state is antisymmetrized in in respect to all the pairwise permutations. The compact representation us by the determinant
$$|\psi_1\psi_2\psi_3\rangle\leftarrow
\frac{1}{n!}\left|\begin{matrix}
\psi_1(x_1)&\psi_2(x_1)&\psi_3(x_1)\\
\psi_1(x_2)&\psi_2(x_2)&\psi_3(x_2)\\
\psi_1(x_3)&\psi_2(x_3)&\psi_3(x_3)
\end{matrix}\right|.$$
Note that the bra-ket notation above is not the same as the bra-kets given in the question, rather corresponds to the Fock space (second quantized representation).
If we really want to construct the many-particles states from the original bras and kets, as a direct product, we need to supplement them with indices indicating which particle occupies the orbital, e.g., we could use notation $$|\psi_i\rangle \rightarrow |\psi_i\rangle_j.$$ Then the construction proceeds exactly as described above, e.g., two particles states can be written as
$$|\psi_{i,j}\rangle_{1,2}=\frac{1}{\sqrt{2}}\left(|\psi_i\rangle_1\otimes |\psi_j\rangle_2 - |\psi_j\rangle_1\otimes |\psi_i\rangle_2\right).$$
