In the following sentence:
高考那天，我紧张极了。
I understand the meaning. But I don't understand why 那天 is used in such a way.
So why is it 高考那天, not something like 高考之天 or 高考的天 (the "of" equivalent in Mandarin)? Or 那高考的天? Is there any difference if you say it as 高考那天 or 高考的天?
The key is that you should understand the English word "OF". OF itself could mean differently in English according to the context.
In this case, I think the reason is that 高考（exam） can not owner a day logically. Similarly in English, the exam's day is logically incorrect. However, 'OF' in English could be used when referring to the day, moment when something happened. For example, the day of the accident. So, in "the day of the exam", it does not express the ownership, but the exam happened in that day or the day in which the exam happened.
Unlike English, we don't have "OF" phrase. So, in Chinese, to interpret "the day of the exam" is equally to interpret "the exam happened in that day" or "the day in which the exam happened". We can use the prep phrase 在...那天，and then it will become 在高考那天. If you omit 在, it will become 高考那天.
On the other hand, OF could be interpreted differently in Chinese according to the context. It could be ...的 denoting the ownership or a prep phrase. For example, "the length of the bench": 凳子的长度. a cup of tea: 一杯茶（you shouldn't translate this to 一杯的茶）.
In summary, there is not a dedicated equivalent in Chinese for the word "OF". You have to make it differently according to the context.
天 in 那天 refers to day, so 高考那天 means the day of 高考.
天 could refer to sky too, such as 高考的天 (and 高考之天), which means the sky of 高考 and doesn't make sense. (It's same for 那高考的天, which could be literally translated as that sky of 高考.)
天 the context is important.
"那天" is used to stress the importance of that day and makes readers to concentrate on that certain day in this particular sentence. When this sentence is said as "那天高考",it aims at stressing what happened at that certain day.
First you should know the mean of 天 could be day or sky as @songyuanyao's answer.
And then, when we take 天 as day, 那天 could be translated as:
When we take 天 as sky, 那天 just means the sky. For example, 那山那人那天那地 means that hill that guy the sky the land. To be frank, we seldom use 那天 to express the sky. It sounds queer.
We seldom use 天 separately to express day. So 高考之天 or 高考的天 seemed stranged to us too.天 usually be used with Quantifier/Article such as 一天/那天. When you want to used a single world to express day, 日 would be a good choice as follows:高考之日/结婚之日/胜利之日.
那高考的天 is somewhat alike to 高考的天, they are not exactly correct in collocation. Even if people dont take this 天 as sky because of the context 高考, people will still be confused when seeing these expression.
Forgive me for my pool english ^_^
高考的天：
Gramatically 的 can be used in different situation. Here it is in the form: noun(高考）+ 的 + noun(天). However, in this form, the second noun should be belongs to the first noun. For example: 圖畫的顏色。天 is not belonging to 高考, therefore the sentence "高考的天" must be wrong. It is not correct even in gramatical sense.
高考之天：
之 is not a modern word when we want to say "of" in English. We seldom say this word in daily conversation unless we want to quote traditional phrases. In ancient time, when we refer to a particular day, "日" is the right word. Therefore it is fine to say 高考之日, because we are using it in an old way. However we cannot say 高考之天，because "天" can only be refer to day in modern Chinese.
高考那天
In fact we can also say 那天高考，but the meaning is a little bit different. The focus of 高考那天 is the day. For example 高考那天不斷下雨。 on the other hands the focus of 那天高考 is the examination. For example 那天高考很容易。
高考之天
is grammatically correct. 之 is more often used in fixed phrases in modern chinese. If we are writing some classical texts, we might use "会考之日". You can say 高考之日 to get an old style.
高考的天
sounds like a day that is good to 高考, or make you have a sense of 高考(高考般的天). This is a bad expression.
高考那天
The day that one attand an examination.
