Is there a proper way of saying you are sorry?

Saying sorry is not commonly taught to us growing up...well, at the very least teaching it is usually done by only giving enough time & thought to just being able to "say" the words, "I'm Sorry".

But that should be enough, right?  I mean, what more than saying the words: "I'm Sorry" needs to be done except perhaps a hug, especially if it is one of your siblings.

Some say that apologizing is both an Art and a Science.  The way you deliver it needs careful consideration as well as preparing the recipe that forms the apology itself must be done with preciseness. 

Apparently there are several "formulas" to the make up of an apology but for time and simplicity sake we are going to focus on only 3 steps.

These are the 3 steps we take here in our family (not repeating the offense is taught but is not necessarily used as a step):
1.  Apologize for the situation:
    "I'm sorry for not sharing my toy with you."
2.  Acknowledge your part in the damage done:
    "It was very selfish of me and I'm sorry that you are feeling sad."
3.  Offer retribution for what you did wrong:
    "Is there anything I can do to make it up to you?"

Now for the Art of Delivery...I believe that is all going to depend on the person delivering the apology.  Since sincerity comes from the heart there will be a greater variety of differences in this area.  Age, even those dealing with emotional disabilities, will need to be taken into account with this as well.

And what about the other person, the one who is being apologized to?  What do they do?  Do they also have some sort of role or part in the processing of an apology?

We think so.  We feel that the person being apologized to needs to know how to handle their end of the exchange. 

For example:  when being apologized to, one must know how to accept the apology.  Is saying that it's okay appropriate?  It may very well some.  But for us, it's like giving them a "get out of jail free" pass.
If it's not okay for what they did wrong, then we shouldn't be telling them it's okay...sends the wrong message.

Then what do you say?
Well, this is sort of the conversation that takes place over here:

Person 1:  "I'm sorry for taking the toy out of your hands. 
                It was very rude of me and I can see that you are sad and upset
                with me."

Person 2:  "Thank you."  or "I forgive you."  or both
               "Thank you, I forgive you."

Person 1:  "What can I do to make things better?"

Person 2:  "I would like for you to use your words next time you
                want something I have.  And, I could really use a hug right

Granted this takes lots and lots of practice and coaching from us parents.  But there are times when we have seen the kids "helping" to teaching or remind each other of these steps...even so, we are often reminded of the steps when we forget.  It definitely is a group/team/family effort.  But in the end...every one feels much better.  Feelings are acknowledged and a plan of action in how to handle the situation next time is put into place.

So, my question to you still do you say you're sorry?