The reason you have awkward eye contacts with her could be that you like her or you notice something interesting and special about her.
Rather than going on with the awkward eye contacts, it's best you man up and have a conversation with her. You can give her a compliment or become friend with her, this will help decrease the awkward eye contacts.
If you like someone, chance are, you will be gaping at him or her every chance you get. After some time, the person either notices or some of her friends tell her about you staring. Naturally, she gazes back and you two make eye contact. Awkward! Well, if you like someone and keep staring at them, you will end up meeting her eyes more than once.
These awkward eye contacts are natural and are usually the start of a conversation. You need to man up and talk to her now. ask her about herself, complement her or think of some work that you might do together. This will help you convert your awkward eye contacts into productive conversation.