The short answer is yes, you do need their written permission.
If you are going abroad, say, for a two-week holiday, then you need to get permission from everyone with Parental Responsibility for the child before taking them abroad. A letter from your ex is usually enough to show you have got that permission.
You can only take your child abroad without getting permission (for up to 28 days) if you have a child arrangement order that says your child must live with you.
If your ex is not willing to give permission, you must apply for a court order to obtain permission to go abroad.
Bear in mind that the family court now places much more emphasis on parents reaching their own agreements and, in most cases, there is a requirement that parents have, at the very least, had a Mediation Information and Assessment Meeting before an application to court can be made.
If your intention is to leave the country for longer or more permanently, and you are unable to agree this with your ex, then you will need to make an application to court for a Specific Issue order.