Will I still be able to use my insurance to drive in other EU member states after Brexit?

Yes. All UK motor insurance providers will continue to provide the legal minimum motor insurance cover for travel to European Economic Area countries ((EEA - all EU countries, plus Norway, Iceland, Liechtenstein). 

If you are a UK motorist, you will, therefore, not need to purchase additional third-party motor insurance policy cover when travelling to these countries with a UK-registered vehicle. You would continue to hold the same third-party cover that you do now.