Why not give this a try?
select{
background: url('data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-chevron-down" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z"/></svg>') no-repeat right #ddd;
-webkit-appearance: none;
padding-right: 1.5rem;
background-position: right 0.1rem center;
}
select:focus {
background: url('data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-chevron-up" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M7.646 4.646a.5.5 0 0 1 .708 0l6 6a.5.5 0 0 1-.708.708L8 5.707l-5.646 5.647a.5.5 0 0 1-.708-.708l6-6z"/></svg>') no-repeat right #ddd;
background-position: right 0.1rem center;
}
<select class="form-select" aria-label="Default select example">
<option selected>Open this select menu</option>
<option value="1">One</option>
<option value="2">Two</option>
<option value="3">Three</option>
</select>
The icons used here are from bootstrap library. You can find them at Chevron up and Chevron down