You're in the perfect position now to set up a regular meeting schedule with your superiors. This will ensure open communication and help both you and your bosses understand and agree on the focus of your job and your performance.
Begin by thoroughly examining your job description to identify each new responsibility and task. Make sure you understand how your superiors want you to deal with issues and what they consider the top three performance criteria.
Since you're new to management, it will be important to listen carefully and learn from those with more experience. Be open minded about what they say and then present your ideas. If your ideas make sense, they'll recognize the validity of your point of view and eventually their trust in you will grow.
Be patient. The time they take to integrate you as the new head of the department doesn't depend on your competencies alone. It often takes longer than expected for everyone to be comfortable with someone new to a management role. It can be easy to assume that your opinion isn't considered in this new hire, but perhaps you'd be wiser to assume that your superiors are taking the time to help you get the right person. Try to turn your frustration into something positive. Instead of wondering about the scope of your responsibilities at this early stage, show your superiors that you're part of the team and give them additional examples of your sound judgment. If you do, your role will broaden naturally over time.