glsl-rotation-2d

vec2 rotate(vec2 v, float a) {
	float s = sin(a);
	float c = cos(a);
	mat2 m = mat2(c, s, -s, c);
	return m * v;
}

glsl-rotation-3d

mat4 rotationMatrix(vec3 axis, float angle) {
    axis = normalize(axis);
    float s = sin(angle);
    float c = cos(angle);
    float oc = 1.0 - c;
    
    return mat4(oc * axis.x * axis.x + c,           oc * axis.x * axis.y - axis.z * s,  oc * axis.z * axis.x + axis.y * s,  0.0,
                oc * axis.x * axis.y + axis.z * s,  oc * axis.y * axis.y + c,           oc * axis.y * axis.z - axis.x * s,  0.0,
                oc * axis.z * axis.x - axis.y * s,  oc * axis.y * axis.z + axis.x * s,  oc * axis.z * axis.z + c,           0.0,
                0.0,                                0.0,                                0.0,                                1.0);
}

vec3 rotate(vec3 v, vec3 axis, float angle) {
	mat4 m = rotationMatrix(axis, angle);
	return (m * vec4(v, 1.0)).xyz;
}

Why does it need to be a mat4?

I'm not sure whether or not it needs to be a mat4. It is probably a mat4 because we're dealing with homogeneous coordinates. Homogeneous coordinates unlock translations under matrix multiplication, as well as projective transformations. To rotate something we first translate it to the rotation axis, apply the rotation, and then translate it back. This is called a similarity transform.