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I don't think we actually need to be able to determine if an electron is spin-up or spin-down; we just need to know +1/2 is spin up and -1/2 is spin down and that two electrons in the same orbital can't have the same spin state.
The convention is to write that electrons fill spin up first. Make sure that you keep the spin consistent until you have filled all the orbitals with one electron (e.g. in the 3 orbitals of the p subshell, fill them up, up, up, down, down, down and not something like up, down, up --> down, up, down). A positive spin is also associated with counterclockwise rotation, while a negative spin is clockwise rotation.
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