Hiding the Arrow of a Drop Down List on exit



I have a form with a drop down list containing various items. Once an item has been selected i want the arrow head to disappear and then reappear on entry again. I believe the only way to do this is to make the field read only on exit and then open again on entry but i have no idea how to write that script. (still learning)!  any help would be appreciated. 

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Did you click it? It worked for me. Only when clicking the dropDownList did it show the arrow, you can select a different item and then it was made readOnly again. The arrow wont show when hovering over it.