I have seen quite a bit online about how there are only 20 independent components for the (lowered) Riemann tensor $R_{abcd}$ for the Schwarzschild metric. I've been told this follows from the symmetries of the tensor, i.e.:
$R_{abcd}=-R_{bacd}=-R_{abdc}=R_{badc}$ and $R_{abcd}=R_{cdab}$
Now if the indices in the tensor can all run from 1 to 4, then $R_{abcd}$ has 256 components. These symmetries seem to reduce our need to calculate components, but why only 20?
Note: I am aware there are some similar questions on the stack exchange. I have read these, but none of them very clearly explained this specific point, so I decided to ask it directly so hopefully I can get my head around it.