defining equation of general projective line Let $\mathbb{P}$ be a projective $n$-space.
For $p=[a_0,\cdots,a_n], q=[b_0,\cdots,b_n]$ I know that the line pass through $p$ and $q$ is defined by the set $\{ [xa_0+yb_0,\cdots, xa_n+yb_n] | [x,y] \in \mathbb{P}^1\}$
I wonder the defining equation of this projective line. and can we formulate the defining equation of the line passing through two point?

The situation isn't really any different from the affine case: the equation of a line in $n$ dimensions is a full rank system of $n-1$ linear equations in $n$ variables.
Homogenizing, in the projective case it is a full rank system of $n-1$ homogeneous linear equations in $n+1$ variables.
As a quick sanity check, the solution to such a thing is two-dimensional. And we have two degrees of freedom: one to choose a point on the line, and one to rescale the coordinates.
To find the system of equations, you are asking to find the matrix $A$ of rank $n-1$ satisfying
$$A \left( \begin{matrix}a_0 & b_0 \\ a_1 & b_1 \\ \vdots & \vdots \\ a_n & b_n \end{matrix} \right) = 0$$
That is, you are simply asking to do a left null space matrix calculation.
