Solution for modern browsers and IE11:
html:
<select class="dropdown-styled">
<option value="1">options 1</option>
<option value="2">options 2</option>
<option value="3">options 3</option>
</select>
<div class="dropdown-styled-wrap">
<select>
<option value="1">options 1</option>
<option value="2">options 2</option>
<option value="3">options 3</option>
</select>
</div>
css:
.dropdown-styled {
appearance: none;
border: 1px solid #555;
width: 80%;
height: 34px;
background: #eee url(https://via.placeholder.com/40x40) right center no-repeat;
}
.dropdown-styled-wrap {
width: 80%;
height: 34px;
overflow: hidden;
background: #eee url(https://via.placeholder.com/40x40) right center no-repeat;
border: 1px solid #555;
}
.dropdown-styled-wrap select {
appearance: none;
background: transparent;
width: 110%;
height: 34px;
border: 0;
}