It's not exactly a vector...
So it can be written as a vector? No?
That's right. It's not a vector because it doesn't "transform" like a vector.
If you take a vector and rotate about an axis by 360 degrees, you get the same vector.
If you take a spinor and rotate it by 360 degrees you get a spinor which is "flipped". You have to rotate the spinor by 720 degrees to get back to the same spinor.
This is intrinsically weird, but that's QM.