Remove Select arrow on IE

In IE9, it is possible with purely a hack as advised by @Spudley. Since you've customized height and width of the div and select, you need to change div:before css to match yours.

In case if it is IE10 then using below css3 it is possible

select::-ms-expand {
    display: none;
}

However if you're interested in jQuery plugin, try Chosen.js or you can create your own in js.