Find the smallest element greater than current element
A better solution would be using a balanced binary search tree.
You can process the elements from the right to left and for each element you find the vertex in the tree with the smallest key greater than this element (a search query on the tree suffice) and then you add the element to the tree.
Total complexity is $O(n\log(n))$ since we are iterating over the elements once and having one query and one insert operations in each iteration.