No, we cannot determine the exact spin of an electron because we cant really isolate or separate an electron to study it. We can only study its effects.
Each orbital can have 2 electrons. If we talk about the p shell, it has three orbitals, Px, Py and Pz. Each of these orbitals contain two electrons making 6 electrons in p shell. Among these two electrons, one spins in clockwise direction whereas another one spins in anti clockwise direction.
But we cannot determine the exact spin so in theory like valence bond theory if there is only one electron in an orbital we assume it in clockwise direction (by giving an arrow pointing upwards)