How to express it? When there is a sequence $a_1, a_2, ..., a_{n-1}, a_n$, I want to express the (n-2) sequence that $a_i, a_j$ are exclude in the previous sequence and inlcude the other remaining elements where $1\le i,j\le n$ and $i\neq j$. But i don't know how to express it formally. 
For example, when i write $a_1, a_i, ..., a_j, a_n$, I concern that when $i=1$ and $j=n$, it seems that the expression means the sequence that $a_1, a_1, a_2, ..., a_n, a_n$ rather than the sequence I intended.

I don't think there is an established easy way to do this. Here are several possibilities:
$$
a_1, \ldots, \hat {a_i}, \ldots, \hat {a_j}, \ldots, a_n
$$
explaining what the hat means, or
$$
a_1, \ldots,   a_{i-1} , a_{i+1}\ldots, a_{j-1} , a_{j+1}, \ldots, a_n
$$
You'd have to trust your reader to make the right sense of this when $i=1$ or $j=n$.
Or just use words: 
$a_1, \ldots, a_n$ omitting $a_i$ and $a_j$.
If you have to do this a lot you could invent a notation, defining (say) $A<i,j>$ to be the sequence $A$ without its $i$th and $j$th elements.
