It is not enough to have conversational skills to become a capable Customer Service Rep. There are other skills necessary in a good Customer Service Rep. Sometimes, people have these skills as part of their personality so they need no further training but in others it may be necessary to go through a company training program to get oriented in appropriate customer service skills.
First, it is necessary to be well-acquainted with what your organization really sells or is offering. A Customer Service Rep who has no idea of what these offerings are will not be able to help anyone. This is why there are seminars, conferences, and training programs that all new recruits have to undergo before becoming full-time Customer Service Reps. This also helps introduce the recruits to how the organization works and who they have to answer to.
Second, you have to have a business-minded personality which means you should know how to be a professional before the clients. Sometimes, the organization has a specific corporate culture so be aware of this to see if you can function as a professional by the norms of that corporate culture. If you cannot, then you might have to seek another job in an organization that is more to your liking.
Third, many organizations may require you to follow a scripted conversation when dealing with clients. One reason for this is that the organization may be fearful of lawsuits so they want to make sure they know what each Customer Service Rep is saying to the customer. Although some people may feel this cramps their customer service style, for others it may seem to be a relief because they need not think up what to say in the conversation and just follow the script. You will have to check whether your organization permits small talk between the client and you.