In trying to begin to learn basic homological algebra, i am confronted with orientation of simplices. The definition seems unmotivated and unintuitive: for $n$-simplices with $n \in \{-1,0,1,2\}$, it is intuitive to visualize: intuitively, given a 2-simplex, we can walk to the left or to the right. For $n=3$ and above, the definition seems arbitrary. Is there a "geometric intuition" for this?
Definition of orientation: two (ordered) simplices with same vertices have same orientation if one is the componentwise image of an even permutation of the other. Else they are said to have opposite orientation.