Consider the group $SO(3)$. The rotation around the $x$-axis is represented by the matrix
$$R_x = \left ( \begin{array}{ ccc } 1 & 0 & 0 \\ 0 & \cos \Theta & - \sin \Theta \\ 0 & \sin \Theta & \cos \Theta \end{array}\right ) $$
Similarly, for the rotations around the other axes. It is very easy to verify that these three are linearly independent.
I wanted to show that they form a basis for $SO(3)$. But this is where I got stuck. The idea should be to consider an arbitrary rotation and then show that it is a linear combination of these $3$ matrices. But I don't know what a generic rotation matrix looks like. So maybe there is a different way of showing this or maybe it is even false.
Please can someone help me?
Basis is the wrong word. However, you are correct; an oriented (orthogonal) basis can be rotated to the standard $x,y,z$ directions in no more than 3 standard rotations. One to put the $x$ vector in the $xz$ plane. a second rotation to send it to the positive $x$ axis. One final rotation in the $yz$ plane to take those vectors to their proper positions.