This is too long for a comment...
Your first statements are not always correct. For three or more particles, one must proceed with extreme care because there are presentations of $S_3$ or $S_n$ that are not 1-dimensional, and neither symmetric nor antisymmetric: a state can be partially symmetric and not return to a multiple of itself after every permutation.
To construct antisymmetric states, one must combine the multidimensional spatial and spin states of representations that transform by conjugate representations of $S_n$. For $S_2$ is collapses to symmetric and antisymmetric representations, respectively but not for $n>3$ or greater. See for instance
Harvey M. The symmetric group and its relevance to fermion physics. CM-P00067619; 1981.
In particular, it is clearly not possible to create fully antisymmetric states of three spin-1/2 particles, so in such a case multidimensional irreps of $S_3$ (or more generally of $S_n$) will have to be combined. In particular the spin and space parts of the states are no longer separable.
This is a very inefficient way, and one usually prefers Slater determinants of space and spin states, which are automatically antisymmetric, as a construction method for fermionic states.
Edit:
In the case of two particles you have in mind, it is possible to separate the spatial and spin degrees of freedom. Thus:
\begin{align}
\Psi_\pm (x_1,x_2)=\left(\phi_1(x_1)\phi_2(x_2)\pm \phi_2(x_1)\phi_1(x_2)\right)\left(\vert +\rangle_1\vert -\rangle_2
\mp \vert -\rangle_1\vert +\rangle_2\right)
\end{align}
are examples of a fully antisymmetric states with total energy $E_1+E_2$
One can also write (for instance) \begin{align} &\hbox{Det}\left\vert \begin{array}{cc} \phi_1(x_1)\vert +\rangle_1 &\phi_2(x_1)\vert -\rangle_1 \\ \phi_1(x_2)\vert +\rangle_2 &\phi_2(x_2)\vert -\rangle_2 \\ \end{array}\right\vert = \phi_1(x_1)\phi_2(x_2)\vert +\rangle_1\vert -\rangle_2 - \phi_1(x_2)\phi_2(x_1)\vert -\rangle_1\vert +\rangle_2\, ,\\ &\hbox{Det}\left\vert \begin{array}{cc} \phi_1(x_1)\vert -\rangle_1 &\phi_2(x_1)\vert +\rangle_1 \\ \phi_1(x_2)\vert -\rangle_2 &\phi_2(x_2)\vert +\rangle_2 \\ \end{array}\right\vert = \phi_1(x_1)\phi_2(x_2)\vert -\rangle_1\vert +\rangle_2 - \phi_1(x_2)\phi_2(x_1)\vert +\rangle_1\vert -\rangle_2\, ,\\ &\hbox{Det}\left\vert \begin{array}{cc} \phi_1(x_1)\vert +\rangle_1 &\phi_2(x_1)\vert +\rangle_1 \\ \phi_1(x_2)\vert +\rangle_2 &\phi_2(x_2)\vert +\rangle_2 \\ \end{array}\right\vert = \left(\phi_1(x_1)\phi_2(x_2)-\phi_2(x_1)\phi_1(x_2)\right)\vert +\rangle_1\vert +\rangle_2\\ &\hbox{Det}\left\vert \begin{array}{cc} \phi_1(x_1)\vert -\rangle_1 &\phi_2(x_1)\vert -\rangle_1 \\ \phi_1(x_2)\vert -\rangle_2 &\phi_2(x_2)\vert -\rangle_2 \\ \end{array}\right\vert = \left(\phi_1(x_1)\phi_2(x_2)-\phi_2(x_1)\phi_1(x_2)\right)\vert -\rangle_1\vert -\rangle_2\end{align} all of which are properly antisymmetrized with total energy $E_1+E_2$.
Note that any linear combination of antisymmetric states is also an antisymmetric states, so additional combinations are possible.